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Arithmetic Sequences

# 92 arithmetic sequences

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Arithmetic Sequences

A sequence a1, a2 , a3 , … is an arithmetic sequence if an = d*n + c, i.e. it is defined by a linear formula.

Arithmetic Sequences

A sequence a1, a2 , a3 , … is an arithmetic sequence if an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbers a1= 1, a2= 3, a3= 5, a4= 7, … is an arithmetic sequence because an = 2n – 1.

Arithmetic Sequences

A sequence a1, a2 , a3 , … is an arithmetic sequence if an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbers a1= 1, a2= 3, a3= 5, a4= 7, … is an arithmetic sequence because an = 2n – 1.Fact: If a1, a2 , a3 , …is an arithmetic sequence and that an = d*n + c then the difference between any two terms is d, i.e. ak+1 – ak = d.

Arithmetic Sequences

A sequence a1, a2 , a3 , … is an arithmetic sequence if an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbers a1= 1, a2= 3, a3= 5, a4= 7, … is an arithmetic sequence because an = 2n – 1.Fact: If a1, a2 , a3 , …is an arithmetic sequence and that an = d*n + c then the difference between any two terms is d, i.e. ak+1 – ak = d.

Arithmetic Sequences

In example A, 3 – 1 = 5 – 3 = 7 – 5 = 2 = d.

A sequence a1, a2 , a3 , … is an arithmetic sequence if an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbers a1= 1, a2= 3, a3= 5, a4= 7, … is an arithmetic sequence because an = 2n – 1.Fact: If a1, a2 , a3 , …is an arithmetic sequence and that an = d*n + c then the difference between any two terms is d, i.e. ak+1 – ak = d.

Arithmetic Sequences

The following theorem gives the converse of the above factand the main formula for arithmetic sequences.

In example A, 3 – 1 = 5 – 3 = 7 – 5 = 2 = d.

A sequence a1, a2 , a3 , … is an arithmetic sequence if an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbers a1= 1, a2= 3, a3= 5, a4= 7, … is an arithmetic sequence because an = 2n – 1.Fact: If a1, a2 , a3 , …is an arithmetic sequence and that an = d*n + c then the difference between any two terms is d, i.e. ak+1 – ak = d.

Arithmetic Sequences

Theorem: If a1, a2 , a3 , …an is a sequence such that an+1 – an = d for all n, then a1, a2, a3,… is an arithmetic sequence

The following theorem gives the converse of the above factand the main formula for arithmetic sequences.

In example A, 3 – 1 = 5 – 3 = 7 – 5 = 2 = d.

A sequence a1, a2 , a3 , … is an arithmetic sequence if an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbers a1= 1, a2= 3, a3= 5, a4= 7, … is an arithmetic sequence because an = 2n – 1.Fact: If a1, a2 , a3 , …is an arithmetic sequence and that an = d*n + c then the difference between any two terms is d, i.e. ak+1 – ak = d.

Arithmetic Sequences

Theorem: If a1, a2 , a3 , …an is a sequence such that an+1 – an = d for all n, then a1, a2, a3,… is an arithmetic sequence and the formula for the sequence is an = d(n – 1) + a1.

The following theorem gives the converse of the above factand the main formula for arithmetic sequences.

In example A, 3 – 1 = 5 – 3 = 7 – 5 = 2 = d.

A sequence a1, a2 , a3 , … is an arithmetic sequence if an = d*n + c, i.e. it is defined by a linear formula.Example A. The sequence of odd numbers a1= 1, a2= 3, a3= 5, a4= 7, … is an arithmetic sequence because an = 2n – 1.Fact: If a1, a2 , a3 , …is an arithmetic sequence and that an = d*n + c then the difference between any two terms is d, i.e. ak+1 – ak = d.

Arithmetic Sequences

Theorem: If a1, a2 , a3 , …an is a sequence such that an+1 – an = d for all n, then a1, a2, a3,… is an arithmetic sequence and the formula for the sequence is an = d(n – 1) + a1. This is the general formula of arithemetic sequences.

The following theorem gives the converse of the above factand the main formula for arithmetic sequences.

In example A, 3 – 1 = 5 – 3 = 7 – 5 = 2 = d.

Arithmetic Sequences Given the description of a arithmetic sequence, we use the general formula to find the specific formula for that sequence.

Example B. Given the sequence 2, 5, 8, 11, … a. Verify it is an arithmetic sequence.

Arithmetic Sequences Given the description of a arithmetic sequence, we use the general formula to find the specific formula for that sequence.

Example B. Given the sequence 2, 5, 8, 11, … a. Verify it is an arithmetic sequence.

It's arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.

Arithmetic Sequences Given the description of a arithmetic sequence, we use the general formula to find the specific formula for that sequence.

Example B. Given the sequence 2, 5, 8, 11, … a. Verify it is an arithmetic sequence.

It's arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.

b. Find the (specific) formula represents this sequence.

Arithmetic Sequences Given the description of a arithmetic sequence, we use the general formula to find the specific formula for that sequence.

Example B. Given the sequence 2, 5, 8, 11, … a. Verify it is an arithmetic sequence.

It's arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.

b. Find the (specific) formula represents this sequence.

Plug a1 = 2 and d = 3, into the general formula an = d(n – 1) + a1

Arithmetic Sequences Given the description of a arithmetic sequence, we use the general formula to find the specific formula for that sequence.

Example B. Given the sequence 2, 5, 8, 11, … a. Verify it is an arithmetic sequence.

It's arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.

b. Find the (specific) formula represents this sequence.

Plug a1 = 2 and d = 3, into the general formula an = d(n – 1) + a1

we get an = 3(n – 1) + 2

Arithmetic Sequences Given the description of a arithmetic sequence, we use the general formula to find the specific formula for that sequence.

Example B. Given the sequence 2, 5, 8, 11, … a. Verify it is an arithmetic sequence.

It's arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.

b. Find the (specific) formula represents this sequence.

Plug a1 = 2 and d = 3, into the general formula an = d(n – 1) + a1

we get an = 3(n – 1) + 2 an = 3n – 3 + 2

Arithmetic Sequences Given the description of a arithmetic sequence, we use the general formula to find the specific formula for that sequence.

Example B. Given the sequence 2, 5, 8, 11, … a. Verify it is an arithmetic sequence.

It's arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.

b. Find the (specific) formula represents this sequence.

Plug a1 = 2 and d = 3, into the general formula an = d(n – 1) + a1

we get an = 3(n – 1) + 2 an = 3n – 3 + 2 an = 3n – 1 the specific formula.

Arithmetic Sequences Given the description of a arithmetic sequence, we use the general formula to find the specific formula for that sequence.

Example B. Given the sequence 2, 5, 8, 11, … a. Verify it is an arithmetic sequence.

It's arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.

b. Find the (specific) formula represents this sequence.

Plug a1 = 2 and d = 3, into the general formula an = d(n – 1) + a1

we get an = 3(n – 1) + 2 an = 3n – 3 + 2 an = 3n – 1 the specific formula.

c. Find a1000.

Arithmetic Sequences Given the description of a arithmetic sequence, we use the general formula to find the specific formula for that sequence.

Example B. Given the sequence 2, 5, 8, 11, … a. Verify it is an arithmetic sequence.

It's arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.

b. Find the (specific) formula represents this sequence.

Plug a1 = 2 and d = 3, into the general formula an = d(n – 1) + a1

we get an = 3(n – 1) + 2 an = 3n – 3 + 2 an = 3n – 1 the specific formula.

c. Find a1000. Set n = 1000 in the specific formula,

Arithmetic Sequences Given the description of a arithmetic sequence, we use the general formula to find the specific formula for that sequence.

Example B. Given the sequence 2, 5, 8, 11, … a. Verify it is an arithmetic sequence.

It's arithmetic because 5 – 2 = 8 – 5 = 11 – 8 = … = 3 = d.

b. Find the (specific) formula represents this sequence.

Plug a1 = 2 and d = 3, into the general formula an = d(n – 1) + a1

we get an = 3(n – 1) + 2 an = 3n – 3 + 2 an = 3n – 1 the specific formula.

c. Find a1000. Set n = 1000 in the specific formula, we get a1000 = 3(1000) – 1 = 2999.

Arithmetic Sequences Given the description of a arithmetic sequence, we use the general formula to find the specific formula for that sequence.

Arithmetic Sequences To use the arithmetic general formula to find the specific formula, we need the first term a1 and the difference d.

Arithmetic Sequences To use the arithmetic general formula to find the specific formula, we need the first term a1 and the difference d.Example C. Given a1, a2 , a3 , …an arithmetic sequence with d = -4 and a6 = 5, find a1, the specific formula and a1000.

Arithmetic Sequences To use the arithmetic general formula to find the specific formula, we need the first term a1 and the difference d.

Set d = –4 in the general formula an = d(n – 1) + a1,

Example C. Given a1, a2 , a3 , …an arithmetic sequence with d = -4 and a6 = 5, find a1, the specific formula and a1000.

Arithmetic Sequences To use the arithmetic general formula to find the specific formula, we need the first term a1 and the difference d.

Set d = –4 in the general formula an = d(n – 1) + a1, we get an = –4(n – 1) + a1.

Example C. Given a1, a2 , a3 , …an arithmetic sequence with d = -4 and a6 = 5, find a1, the specific formula and a1000.

Arithmetic Sequences To use the arithmetic general formula to find the specific formula, we need the first term a1 and the difference d.

Set d = –4 in the general formula an = d(n – 1) + a1, we get an = –4(n – 1) + a1. Set n = 6 in this formula,

Example C. Given a1, a2 , a3 , …an arithmetic sequence with d = -4 and a6 = 5, find a1, the specific formula and a1000.

Arithmetic Sequences To use the arithmetic general formula to find the specific formula, we need the first term a1 and the difference d.

Set d = –4 in the general formula an = d(n – 1) + a1, we get an = –4(n – 1) + a1. Set n = 6 in this formula, we geta6 = -4(6 – 1) + a1 = 5

Example C. Given a1, a2 , a3 , …an arithmetic sequence with d = -4 and a6 = 5, find a1, the specific formula and a1000.

Arithmetic Sequences To use the arithmetic general formula to find the specific formula, we need the first term a1 and the difference d.

Set d = –4 in the general formula an = d(n – 1) + a1, we get an = –4(n – 1) + a1. Set n = 6 in this formula, we geta6 = -4(6 – 1) + a1 = 5 -20 + a1 = 5

Example C. Given a1, a2 , a3 , …an arithmetic sequence with d = -4 and a6 = 5, find a1, the specific formula and a1000.

Arithmetic Sequences To use the arithmetic general formula to find the specific formula, we need the first term a1 and the difference d.

Set d = –4 in the general formula an = d(n – 1) + a1, we get an = –4(n – 1) + a1. Set n = 6 in this formula, we geta6 = -4(6 – 1) + a1 = 5 -20 + a1 = 5 a1 = 25

Example C. Given a1, a2 , a3 , …an arithmetic sequence with d = -4 and a6 = 5, find a1, the specific formula and a1000.

Arithmetic Sequences To use the arithmetic general formula to find the specific formula, we need the first term a1 and the difference d.

Set d = –4 in the general formula an = d(n – 1) + a1, we get an = –4(n – 1) + a1. Set n = 6 in this formula, we geta6 = -4(6 – 1) + a1 = 5 -20 + a1 = 5 a1 = 25To find the specific formula , set 25 for a1 in an = -4(n – 1) + a1

Example C. Given a1, a2 , a3 , …an arithmetic sequence with d = -4 and a6 = 5, find a1, the specific formula and a1000.

Arithmetic Sequences To use the arithmetic general formula to find the specific formula, we need the first term a1 and the difference d.

Set d = –4 in the general formula an = d(n – 1) + a1, we get an = –4(n – 1) + a1. Set n = 6 in this formula, we geta6 = -4(6 – 1) + a1 = 5 -20 + a1 = 5 a1 = 25To find the specific formula , set 25 for a1 in an = -4(n – 1) + a1

an = -4(n – 1) + 25

Example C. Given a1, a2 , a3 , …an arithmetic sequence with d = -4 and a6 = 5, find a1, the specific formula and a1000.

Arithmetic Sequences To use the arithmetic general formula to find the specific formula, we need the first term a1 and the difference d.

Set d = –4 in the general formula an = d(n – 1) + a1, we get an = –4(n – 1) + a1. Set n = 6 in this formula, we geta6 = -4(6 – 1) + a1 = 5 -20 + a1 = 5 a1 = 25To find the specific formula , set 25 for a1 in an = -4(n – 1) + a1

an = -4(n – 1) + 25 an = -4n + 4 + 25

Example C. Given a1, a2 , a3 , …an arithmetic sequence with d = -4 and a6 = 5, find a1, the specific formula and a1000.

Arithmetic Sequences To use the arithmetic general formula to find the specific formula, we need the first term a1 and the difference d.

Set d = –4 in the general formula an = d(n – 1) + a1, we get an = –4(n – 1) + a1. Set n = 6 in this formula, we geta6 = -4(6 – 1) + a1 = 5 -20 + a1 = 5 a1 = 25To find the specific formula , set 25 for a1 in an = -4(n – 1) + a1

an = -4(n – 1) + 25 an = -4n + 4 + 25 an = -4n + 29

Example C. Given a1, a2 , a3 , …an arithmetic sequence with d = -4 and a6 = 5, find a1, the specific formula and a1000.

Arithmetic Sequences To use the arithmetic general formula to find the specific formula, we need the first term a1 and the difference d.

Set d = –4 in the general formula an = d(n – 1) + a1, we get an = –4(n – 1) + a1. Set n = 6 in this formula, we geta6 = -4(6 – 1) + a1 = 5 -20 + a1 = 5 a1 = 25To find the specific formula , set 25 for a1 in an = -4(n – 1) + a1

an = -4(n – 1) + 25 an = -4n + 4 + 25 an = -4n + 29

Example C. Given a1, a2 , a3 , …an arithmetic sequence with d = -4 and a6 = 5, find a1, the specific formula and a1000.

To find a1000, set n = 1000 in the specific formula

Arithmetic Sequences To use the arithmetic general formula to find the specific formula, we need the first term a1 and the difference d.

Set d = –4 in the general formula an = d(n – 1) + a1, we get an = –4(n – 1) + a1. Set n = 6 in this formula, we geta6 = -4(6 – 1) + a1 = 5 -20 + a1 = 5 a1 = 25To find the specific formula , set 25 for a1 in an = -4(n – 1) + a1

an = -4(n – 1) + 25 an = -4n + 4 + 25 an = -4n + 29

Example C. Given a1, a2 , a3 , …an arithmetic sequence with d = -4 and a6 = 5, find a1, the specific formula and a1000.

To find a1000, set n = 1000 in the specific formulaa1000 = –4(1000) + 29 = –3971

Example D. Given that a1, a2 , a3 , …is an arithmetic sequence with a3 = -3 and a9 = 39, find d, a1 and the specific formula.

Arithmetic Sequences

Example D. Given that a1, a2 , a3 , …is an arithmetic sequence with a3 = -3 and a9 = 39, find d, a1 and the specific formula.

Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1,

Arithmetic Sequences

Example D. Given that a1, a2 , a3 , …is an arithmetic sequence with a3 = -3 and a9 = 39, find d, a1 and the specific formula.

Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a3 = d(3 – 1) + a1 = -3

Arithmetic Sequences

Example D. Given that a1, a2 , a3 , …is an arithmetic sequence with a3 = -3 and a9 = 39, find d, a1 and the specific formula.

Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a3 = d(3 – 1) + a1 = -3 2d + a1 = -3

Arithmetic Sequences

Example D. Given that a1, a2 , a3 , …is an arithmetic sequence with a3 = -3 and a9 = 39, find d, a1 and the specific formula.

Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a3 = d(3 – 1) + a1 = -3 2d + a1 = -3

Arithmetic Sequences

a9 = d(9 – 1) + a1 = 39

Example D. Given that a1, a2 , a3 , …is an arithmetic sequence with a3 = -3 and a9 = 39, find d, a1 and the specific formula.

Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a3 = d(3 – 1) + a1 = -3 2d + a1 = -3

Arithmetic Sequences

a9 = d(9 – 1) + a1 = 39 8d + a1 = 39

Example D. Given that a1, a2 , a3 , …is an arithmetic sequence with a3 = -3 and a9 = 39, find d, a1 and the specific formula.

Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a3 = d(3 – 1) + a1 = -3 2d + a1 = -3 Subtract these equations: 8d + a1 = 39 ) 2d + a1 = -3

Arithmetic Sequences

a9 = d(9 – 1) + a1 = 39 8d + a1 = 39

Example D. Given that a1, a2 , a3 , …is an arithmetic sequence with a3 = -3 and a9 = 39, find d, a1 and the specific formula.

Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a3 = d(3 – 1) + a1 = -3 2d + a1 = -3 Subtract these equations: 8d + a1 = 39 ) 2d + a1 = -3 6d = 42

Arithmetic Sequences

a9 = d(9 – 1) + a1 = 39 8d + a1 = 39

Example D. Given that a1, a2 , a3 , …is an arithmetic sequence with a3 = -3 and a9 = 39, find d, a1 and the specific formula.

Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a3 = d(3 – 1) + a1 = -3 2d + a1 = -3 Subtract these equations: 8d + a1 = 39 ) 2d + a1 = -3 6d = 42 d = 7

Arithmetic Sequences

a9 = d(9 – 1) + a1 = 39 8d + a1 = 39

Example D. Given that a1, a2 , a3 , …is an arithmetic sequence with a3 = -3 and a9 = 39, find d, a1 and the specific formula.

Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a3 = d(3 – 1) + a1 = -3 2d + a1 = -3 Subtract these equations: 8d + a1 = 39 ) 2d + a1 = -3 6d = 42 d = 7 Put d = 7 into 2d + a1 = -3,

Arithmetic Sequences

a9 = d(9 – 1) + a1 = 39 8d + a1 = 39

Example D. Given that a1, a2 , a3 , …is an arithmetic sequence with a3 = -3 and a9 = 39, find d, a1 and the specific formula.

Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a3 = d(3 – 1) + a1 = -3 2d + a1 = -3 Subtract these equations: 8d + a1 = 39 ) 2d + a1 = -3 6d = 42 d = 7 Put d = 7 into 2d + a1 = -3, 2(7) + a1 = -3

Arithmetic Sequences

a9 = d(9 – 1) + a1 = 39 8d + a1 = 39

Example D. Given that a1, a2 , a3 , …is an arithmetic sequence with a3 = -3 and a9 = 39, find d, a1 and the specific formula.

Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a3 = d(3 – 1) + a1 = -3 2d + a1 = -3 Subtract these equations: 8d + a1 = 39 ) 2d + a1 = -3 6d = 42 d = 7 Put d = 7 into 2d + a1 = -3, 2(7) + a1 = -3 14 + a1 = -3

Arithmetic Sequences

a9 = d(9 – 1) + a1 = 39 8d + a1 = 39

Example D. Given that a1, a2 , a3 , …is an arithmetic sequence with a3 = -3 and a9 = 39, find d, a1 and the specific formula.

Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a3 = d(3 – 1) + a1 = -3 2d + a1 = -3 Subtract these equations: 8d + a1 = 39 ) 2d + a1 = -3 6d = 42 d = 7 Put d = 7 into 2d + a1 = -3, 2(7) + a1 = -3 14 + a1 = -3 a1 = -17

Arithmetic Sequences

a9 = d(9 – 1) + a1 = 39 8d + a1 = 39

Example D. Given that a1, a2 , a3 , …is an arithmetic sequence with a3 = -3 and a9 = 39, find d, a1 and the specific formula.

Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a3 = d(3 – 1) + a1 = -3 2d + a1 = -3 Subtract these equations: 8d + a1 = 39 ) 2d + a1 = -3 6d = 42 d = 7 Put d = 7 into 2d + a1 = -3, 2(7) + a1 = -3 14 + a1 = -3 a1 = -17Hence the specific formula is an = 7(n – 1) – 17

Arithmetic Sequences

a9 = d(9 – 1) + a1 = 39 8d + a1 = 39

Example D. Given that a1, a2 , a3 , …is an arithmetic sequence with a3 = -3 and a9 = 39, find d, a1 and the specific formula.

Set n = 3 and n = 9 in the general arithmetic formulaan = d(n – 1) + a1, we get a3 = d(3 – 1) + a1 = -3 2d + a1 = -3 Subtract these equations: 8d + a1 = 39 ) 2d + a1 = -3 6d = 42 d = 7 Put d = 7 into 2d + a1 = -3, 2(7) + a1 = -3 14 + a1 = -3 a1 = -17Hence the specific formula is an = 7(n – 1) – 17 or an = 7n – 24.

Arithmetic Sequences

a9 = d(9 – 1) + a1 = 39 8d + a1 = 39

Given that a1, a2 , a3 , …an an arithmetic sequence, then

a1+ a2 + a3 + … + an = n

Sum of Arithmetic Sequences

Given that a1, a2 , a3 , …an an arithmetic sequence, then

a1+ a2 + a3 + … + an = n

Sum of Arithmetic Sequences

Given that a1, a2 , a3 , …an an arithmetic sequence, then

a1+ a2 + a3 + … + an = n

an a1 + 2( )= n

Sum of Arithmetic Sequences

Given that a1, a2 , a3 , …an an arithmetic sequence, then

a1+ a2 + a3 + … + an = n

an a1 + 2( )= n

Example E. a. Given the arithmetic sequence a1= 4, 7, 10, … , and an = 67. What is n?

Sum of Arithmetic Sequences

Given that a1, a2 , a3 , …an an arithmetic sequence, then

a1+ a2 + a3 + … + an = n

an a1 + 2( )= n

Example E. a. Given the arithmetic sequence a1= 4, 7, 10, … , and an = 67. What is n?

We need the specific formula.

Sum of Arithmetic Sequences

Given that a1, a2 , a3 , …an an arithmetic sequence, then

a1+ a2 + a3 + … + an = n

an a1 + 2( )= n

Example E. a. Given the arithmetic sequence a1= 4, 7, 10, … , and an = 67. What is n?

We need the specific formula. Find d = 7 – 4 = 3.

Sum of Arithmetic Sequences

Given that a1, a2 , a3 , …an an arithmetic sequence, then

a1+ a2 + a3 + … + an = n

an a1 + 2( )= n

Example E. a. Given the arithmetic sequence a1= 4, 7, 10, … , and an = 67. What is n?

We need the specific formula. Find d = 7 – 4 = 3. Therefore the specific formula is an = 3(n – 1) + 4

Sum of Arithmetic Sequences

Given that a1, a2 , a3 , …an an arithmetic sequence, then

a1+ a2 + a3 + … + an = n

an a1 + 2( )= n

Example E. a. Given the arithmetic sequence a1= 4, 7, 10, … , and an = 67. What is n?

We need the specific formula. Find d = 7 – 4 = 3. Therefore the specific formula is an = 3(n – 1) + 4 an = 3n + 1.

Sum of Arithmetic Sequences

Given that a1, a2 , a3 , …an an arithmetic sequence, then

a1+ a2 + a3 + … + an = n

an a1 + 2( )= n

Example E. a. Given the arithmetic sequence a1= 4, 7, 10, … , and an = 67. What is n?

We need the specific formula. Find d = 7 – 4 = 3. Therefore the specific formula is an = 3(n – 1) + 4 an = 3n + 1.

Sum of Arithmetic Sequences

If an = 67 = 3n + 1,

Given that a1, a2 , a3 , …an an arithmetic sequence, then

a1+ a2 + a3 + … + an = n

an a1 + 2( )= n

Example E. a. Given the arithmetic sequence a1= 4, 7, 10, … , and an = 67. What is n?

We need the specific formula. Find d = 7 – 4 = 3. Therefore the specific formula is an = 3(n – 1) + 4 an = 3n + 1.

Sum of Arithmetic Sequences

If an = 67 = 3n + 1, then 66 = 3n

Given that a1, a2 , a3 , …an an arithmetic sequence, then

a1+ a2 + a3 + … + an = n

an a1 + 2( )= n

Example E. a. Given the arithmetic sequence a1= 4, 7, 10, … , and an = 67. What is n?

We need the specific formula. Find d = 7 – 4 = 3. Therefore the specific formula is an = 3(n – 1) + 4 an = 3n + 1.

Sum of Arithmetic Sequences

If an = 67 = 3n + 1, then 66 = 3n or 22 = n

b. Find the sum 4 + 7 + 10 +…+ 67

Sum of Arithmetic Sequences

b. Find the sum 4 + 7 + 10 +…+ 67

a1 = 4, and a22 = 67 with n = 22,

Sum of Arithmetic Sequences

b. Find the sum 4 + 7 + 10 +…+ 67

a1 = 4, and a22 = 67 with n = 22, so the sum

4 + 7 + 10 +…+ 67 = 22 4 + 67

2( )

Sum of Arithmetic Sequences

b. Find the sum 4 + 7 + 10 +…+ 67

a1 = 4, and a22 = 67 with n = 22, so the sum

4 + 7 + 10 +…+ 67 = 22 4 + 67

2( )11

Sum of Arithmetic Sequences

b. Find the sum 4 + 7 + 10 +…+ 67

a1 = 4, and a22 = 67 with n = 22, so the sum

4 + 7 + 10 +…+ 67 = 22 = 11(71) 4 + 67

2( )11

Sum of Arithmetic Sequences

b. Find the sum 4 + 7 + 10 +…+ 67

a1 = 4, and a22 = 67 with n = 22, so the sum

4 + 7 + 10 +…+ 67 = 22 = 11(71) = 7814 + 67

2( )11

Sum of Arithmetic Sequences

Arithmetic Sequences

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