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11.2: Arithmetic Sequences & Series. n th Term of an Arithmetic Sequence: a n = a 1 + ( n – 1) d Ex. 1 Determine the following using the table below. n th Term of an Arithmetic Sequence: a n = a 1 + ( n – 1) d Ex. 1 Determine the following using the table below. - PowerPoint PPT Presentation
11.2: Arithmetic Sequences & 11.2: Arithmetic Sequences & SeriesSeries
nnthth Term of an Arithmetic Term of an Arithmetic Sequence:Sequence:
aann = = aa11 + ( + (nn – 1) – 1)dd
Ex. 1Ex. 1Determine the following using the Determine the following using the table below.table below.
5555 4949 4343 3737
aa11 aa22 aa33 aa44
nnthth Term of an Arithmetic Term of an Arithmetic Sequence:Sequence:
aann = = aa11 + ( + (nn – 1) – 1)dd
Ex. 1Ex. 1Determine the following using the Determine the following using the table below.table below.
a) Find the 10a) Find the 10thth term in the sequence. term in the sequence.
5555 4949 4343 3737
aa11 aa22 aa33 aa44
nnthth Term of an Arithmetic Term of an Arithmetic Sequence:Sequence:
aann = = aa11 + ( + (nn – 1) – 1)dd
Ex. 1Ex. 1Determine the following using the Determine the following using the table below.table below.
a) Find the 10a) Find the 10thth term in the sequence. term in the sequence.
aann = = aa11 + ( + (nn – 1) – 1)dd
5555 4949 4343 3737
aa11 aa22 aa33 aa44
nnthth Term of an Arithmetic Term of an Arithmetic Sequence:Sequence:
aann = = aa11 + ( + (nn – 1) – 1)dd
Ex. 1Ex. 1Determine the following using the Determine the following using the table below.table below.
a) Find the a) Find the 1010thth term in the sequence. term in the sequence.
aann = = aa11 + ( + (nn – 1) – 1)dd
aa1010 = = aa11 + ( + (1010 – 1) – 1)dd
5555 4949 4343 3737
aa11 aa22 aa33 aa44
nnthth Term of an Arithmetic Term of an Arithmetic Sequence:Sequence:
aann = = aa11 + ( + (nn – 1) – 1)dd
Ex. 1Ex. 1Determine the following using the Determine the following using the table below.table below.
a) Find the 10a) Find the 10thth term in the sequence. term in the sequence.
aann = = aa11 + ( + (nn – 1) – 1)dd
aa1010 = = 5555 + (10 – 1) + (10 – 1)dd
5555 4949 4343 3737
aa11 aa22 aa33 aa44
nnthth Term of an Arithmetic Term of an Arithmetic Sequence:Sequence:
aann = = aa11 + ( + (nn – 1) – 1)dd
Ex. 1Ex. 1Determine the following using the Determine the following using the table below.table below.
a) Find the 10a) Find the 10thth term in the sequence. term in the sequence.
aann = = aa11 + ( + (nn – 1) – 1)dd
aa1010 = 55 + (10 – 1) = 55 + (10 – 1)(-6)(-6)
5555 4949 4343 3737
aa11 aa22 aa33 aa44
nnthth Term of an Arithmetic Term of an Arithmetic Sequence:Sequence:
aann = = aa11 + ( + (nn – 1) – 1)dd
Ex. 1Ex. 1Determine the following using the Determine the following using the table below.table below.
a) Find the 10a) Find the 10thth term in the sequence. term in the sequence.
aann = = aa11 + ( + (nn – 1) – 1)dd
aa1010 = 55 + (10 – 1)(-6) = 55 + (10 – 1)(-6)
5555 4949 4343 3737
aa11 aa22 aa33 aa44
nnthth Term of an Arithmetic Sequence: Term of an Arithmetic Sequence:
aann = = aa11 + ( + (nn – 1) – 1)dd
Ex. 1Ex. 1Determine the following using the Determine the following using the table below.table below.
a) Find the 10a) Find the 10thth term in the sequence. term in the sequence.
aann = = aa11 + ( + (nn – 1) – 1)dd
aa1010 = 55 + ( = 55 + (10 – 110 – 1)(-6))(-6)
aa1010 = 55 + ( = 55 + (99)(-6))(-6)
5555 4949 4343 3737
aa11 aa22 aa33 aa44
nnthth Term of an Arithmetic Sequence: Term of an Arithmetic Sequence:
aann = = aa11 + ( + (nn – 1) – 1)dd
Ex. 1Ex. 1Determine the following using the Determine the following using the table below.table below.
a) Find the 10a) Find the 10thth term in the sequence. term in the sequence.
aann = = aa11 + ( + (nn – 1) – 1)dd
aa1010 = 55 + (10 – 1)(-6) = 55 + (10 – 1)(-6)
aa1010 = 55 + = 55 + (9)(-6)(9)(-6)
aa1010 = 55 = 55 – 54 – 54
5555 4949 4343 3737
aa11 aa22 aa33 aa44
nnthth Term of an Arithmetic Sequence: Term of an Arithmetic Sequence:
aann = = aa11 + ( + (nn – 1) – 1)dd
Ex. 1Ex. 1Determine the following using the Determine the following using the table table below.below.
a) Find the 10a) Find the 10thth term in the sequence. term in the sequence.
aann = = aa11 + ( + (nn – 1) – 1)dd
aa1010 = 55 + (10 – 1)(-6) = 55 + (10 – 1)(-6)
aa1010 = 55 + (9)(-6) = 55 + (9)(-6)
aa1010 = 55 – 54 = 55 – 54
aa1010 = 1 = 1
5555 4949 4343 3737
aa11 aa22 aa33 aa44
b) Write an equation for the b) Write an equation for the nnthth term term of of the sequence.the sequence.
b) Write an equation for the b) Write an equation for the nnthth term term of of the sequence.the sequence.
aann = = aa11 + ( + (nn – 1) – 1)dd
b) Write an equation for the b) Write an equation for the nnthth term term of of the sequence.the sequence.
aann = = aa11 + ( + (nn – 1) – 1)dd
aann = = 5555 + ( + (nn – 1) – 1)(-6)(-6)
b) Write an equation for the b) Write an equation for the nnthth term term of of the sequence.the sequence.
aann = = aa11 + ( + (nn – 1) – 1)dd
aann = 55 + ( = 55 + (nn – 1) – 1)(-6)(-6)
aann = 55 = 55 - 6- 6((nn – 1) – 1)
b) Write an equation for the b) Write an equation for the nnthth term term of of the sequence.the sequence.
aann = = aa11 + ( + (nn – 1) – 1)dd
aann = 55 + ( = 55 + (nn – 1)(-6) – 1)(-6)
aann = 55 = 55 - 6(- 6(nn – 1) – 1) aann = 55 = 55 - 6- 6nn + 6 + 6
b) Write an equation for the b) Write an equation for the nnthth term term of of the sequence.the sequence.
aann = = aa11 + ( + (nn – 1) – 1)dd
aann = 55 + ( = 55 + (nn – 1)(-6) – 1)(-6)
aann = 55 - 6( = 55 - 6(nn – 1) – 1) aann = = 5555 - 6 - 6nn + 6 + 6
aann = - 6 = - 6nn + 61 + 61
b) Write an equation for the b) Write an equation for the nnthth term term of of the sequence.the sequence.
aann = = aa11 + ( + (nn – 1) – 1)dd
aann = 55 + ( = 55 + (nn – 1)(-6) – 1)(-6)
aann = 55 - 6( = 55 - 6(nn – 1) – 1) aann = 55 - 6 = 55 - 6nn + 6 + 6
aann = - 6 = - 6nn + 61 + 61
Ex. 2Ex. 2 Find the arithmetic means in the Find the arithmetic means in the sequence below.sequence below.
24, ___, ___, ___, ___, -124, ___, ___, ___, ___, -1
Ex. 2Ex. 2 Find the arithmetic means in the Find the arithmetic means in the sequence below.sequence below.
24, ___, ___, ___, ___, -124, ___, ___, ___, ___, -1
***Find the missing terms in the sequence!***Find the missing terms in the sequence!
Ex. 2Ex. 2 Find the arithmetic means in the Find the arithmetic means in the sequence below.sequence below.
24, ___, ___, ___, ___, -124, ___, ___, ___, ___, -1
aa11 aa22 aa33 aa44 aa55 aa66
Ex. 2Ex. 2 Find the arithmetic means in the Find the arithmetic means in the sequence below.sequence below.
2424, ___, ___, ___, ___, , ___, ___, ___, ___, -1-1
aa11 aa22 aa33 aa44 aa55 aa66
Ex. 2Ex. 2 Find the arithmetic means in the Find the arithmetic means in the sequence below.sequence below.
2424, ___, ___, ___, ___, , ___, ___, ___, ___, -1-1
aa11 aa22 aa33 aa44 aa55 aa66
nn = 6 = 6
aa11 = 24 = 24
aa66 = -1 = -1
Ex. 2Ex. 2 Find the arithmetic means in the Find the arithmetic means in the sequence below.sequence below.
2424, ___, ___, ___, ___, , ___, ___, ___, ___, -1-1
aa11 aa22 aa33 aa44 aa55 aa66
nn = 6 = 6
aa11 = 24 = 24
aa66 = -1 = -1
aann = = aa11 + ( + (nn – 1) – 1)dd
Ex. 2Ex. 2 Find the arithmetic means in the Find the arithmetic means in the sequence below.sequence below.
2424, ___, ___, ___, ___, , ___, ___, ___, ___, -1-1
aa11 aa22 aa33 aa44 aa55 aa66
nn = 6 = 6
aa11 = 24 = 24
aa66 = -1 = -1
aann = = aa11 + ( + (nn – 1) – 1)dd
-1-1 = = 2424 + (+ (66 – 1) – 1)dd
Ex. 2Ex. 2 Find the arithmetic means in the Find the arithmetic means in the sequence below.sequence below.
2424, ___, ___, ___, ___, , ___, ___, ___, ___, -1-1
aa11 aa22 aa33 aa44 aa55 aa66
nn = 6 = 6 aa11 = 24 = 24 aa66 = -1 = -1
aann = = aa11 + ( + (nn – 1) – 1)dd
-1-1 = = 2424 + (+ (66 – 1) – 1)dd
-1 = 24 + 5-1 = 24 + 5dd
-25 = 5-25 = 5dd
-5 = -5 = dd
Ex. 2Ex. 2 Find the arithmetic means in the Find the arithmetic means in the sequence below.sequence below.
2424, ___, ___, ___, ___, , ___, ___, ___, ___, -1-1
aa11 aa22 aa33 aa44 aa55 aa66
nn = 6 = 6 aa11 = 24 = 24 aa66 = -1 = -1
aann = = aa11 + ( + (nn – 1) – 1)dd
-1-1 = = 2424 + (+ (66 – 1) – 1)dd
-5 = -5 = dd
aa11 = 24 = 24
Ex. 2Ex. 2 Find the arithmetic means in the Find the arithmetic means in the sequence below.sequence below.
2424, ___, ___, ___, ___, , ___, ___, ___, ___, -1-1
aa11 aa22 aa33 aa44 aa55 aa66
nn = 6 = 6 aa11 = 24 = 24 aa66 = -1 = -1
aann = = aa11 + ( + (nn – 1) – 1)dd
-1-1 = = 2424 + (+ (66 – 1) – 1)dd
-5 = -5 = dd
aa11 = 24 = 24
aa22 = 24 + (-5) = 19 = 24 + (-5) = 19
Ex. 2Ex. 2 Find the arithmetic means in the Find the arithmetic means in the sequence below.sequence below.
2424, ___, ___, ___, ___, , ___, ___, ___, ___, -1-1
aa11 aa22 aa33 aa44 aa55 aa66
nn = 6 = 6 aa11 = 24 = 24 aa66 = -1 = -1
aann = = aa11 + ( + (nn – 1) – 1)dd
-1-1 = = 2424 + (+ (66 – 1) – 1)dd
-5 = -5 = dd
aa11 = 24 = 24
aa22 = 24 + (-5) = 19 = 24 + (-5) = 19
aa33 = 19 + (-5) = 14 = 19 + (-5) = 14
Ex. 2Ex. 2 Find the arithmetic means in the Find the arithmetic means in the sequence below.sequence below.
2424, ___, ___, ___, ___, , ___, ___, ___, ___, -1-1
aa11 aa22 aa33 aa44 aa55 aa66
nn = 6 = 6 aa11 = 24 = 24 aa66 = -1 = -1 dd = -5 = -5
aa11 = 24 = 24
aa22 = 24 + (-5) = 19 = 24 + (-5) = 19
aa33 = 19 + (-5) = 14 = 19 + (-5) = 14
aa44 = 14 + (-5) = 9 = 14 + (-5) = 9
Ex. 2Ex. 2 Find the arithmetic means in the Find the arithmetic means in the sequence below.sequence below.
2424, ___, ___, ___, ___, , ___, ___, ___, ___, -1-1
aa11 aa22 aa33 aa44 aa55 aa66
nn = 6 = 6 aa11 = 24 = 24 aa66 = -1 = -1 dd = -5 = -5
aa11 = 24 = 24
aa22 = 24 + (-5) = 19 = 24 + (-5) = 19
aa33 = 19 + (-5) = 14 = 19 + (-5) = 14
aa44 = 14 + (-5) = 9 = 14 + (-5) = 9
aa55 = 9 + (-5) = 4 = 9 + (-5) = 4
Ex. 2Ex. 2 Find the arithmetic means in the Find the arithmetic means in the sequence below.sequence below.
2424, ___, ___, ___, ___, , ___, ___, ___, ___, -1-1
aa11 aa22 aa33 aa44 aa55 aa66
nn = 6 = 6 aa11 = 24 = 24 aa66 = -1 = -1 dd = -5 = -5
aa11 = 24 = 24
aa22 = 24 + (-5) = = 24 + (-5) = 1919
aa33 = 19 + (-5) = = 19 + (-5) = 1414
aa44 = 14 + (-5) = = 14 + (-5) = 99
aa55 = 9 + (-5) = = 9 + (-5) = 44
Sum of an Arithmetic SeriesSum of an Arithmetic Series
Sum of an Arithmetic SeriesSum of an Arithmetic Series
The sum The sum SSnn of the first of the first nn terms of an terms of an
arithmetic series is given by the following:arithmetic series is given by the following:
Sum of an Arithmetic SeriesSum of an Arithmetic Series
The sum The sum SSnn of the first of the first nn terms of an terms of an
arithmetic series is given by the following:arithmetic series is given by the following:
SSnn = ½ = ½nn[ 2[ 2aa11 + ( + (nn – 1) – 1)d d ]]
Sum of an Arithmetic SeriesSum of an Arithmetic Series
The sum The sum SSnn of the first of the first nn terms of an terms of an
arithmetic series is given by the following:arithmetic series is given by the following:
SSnn = ½ = ½nn[ 2[ 2aa11 + ( + (nn – 1) – 1)d d ]]
OROR SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
Sum of an Arithmetic SeriesSum of an Arithmetic Series
The sum The sum SSnn of the first of the first nn terms of an terms of an
arithmetic series is given by the following:arithmetic series is given by the following:
SSnn = ½ = ½nn[ 2[ 2aa11 + + ((nn – 1) – 1)dd ]]
OROR SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
Sum of an Arithmetic SeriesSum of an Arithmetic Series
The sum The sum SSnn of the first of the first nn terms of an terms of an
arithmetic series is given by the following:arithmetic series is given by the following:
SSnn = ½ = ½nn[ [ 22aa11 + ( + (nn – 1) – 1)d d ]]
OROR SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
Sum of an Arithmetic SeriesSum of an Arithmetic Series
The sum The sum SSnn of the first of the first nn terms of an terms of an
arithmetic series is given by the following:arithmetic series is given by the following:
SSnn = ½ = ½nn[ 2[ 2aa11 + ( + (nn – 1) – 1)d d ]]
OROR SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
Ex. 3Ex. 3 Find Find SSnn for each of the following:for each of the following:
Sum of an Arithmetic SeriesSum of an Arithmetic Series
The sum The sum SSnn of the first of the first nn terms of an terms of an
arithmetic series is given by the following:arithmetic series is given by the following:
SSnn = ½ = ½nn[ 2[ 2aa11 + ( + (nn – 1) – 1)d d ]]
OROR SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
Ex. 3Ex. 3 Find Find SSnn for each of the following:for each of the following:
a) a) aa11 = 58, = 58, aann = -7, = -7, nn = 26 = 26
Sum of an Arithmetic SeriesSum of an Arithmetic Series
The sum The sum SSnn of the first of the first nn terms of an terms of an
arithmetic series is given by the following:arithmetic series is given by the following:
SSnn = ½ = ½nn[ 2[ 2aa11 + ( + (nn – 1) – 1)d d ]]
OROR SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
Ex. 3Ex. 3 Find Find SSnn for each of the following:for each of the following:
a) a) aa11 = 58, = 58, aann = -7, = -7, nn = 26 = 26
Sum of an Arithmetic SeriesSum of an Arithmetic Series
The sum The sum SSnn of the first of the first nn terms of an terms of an
arithmetic series is given by the following:arithmetic series is given by the following:
SSnn = ½ = ½nn[ 2[ 2aa11 + ( + (nn – 1) – 1)d d ]]
OROR SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
Ex. 3Ex. 3 Find Find SSnn for each of the following:for each of the following:
a) a) aa11 = 58, = 58, aann = -7, = -7, nn = 26 = 26
SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
Sum of an Arithmetic SeriesSum of an Arithmetic Series
The sum The sum SSnn of the first of the first nn terms of an terms of an
arithmetic series is given by the following:arithmetic series is given by the following:
SSnn = ½ = ½nn[ 2[ 2aa11 + ( + (nn – 1) – 1)d d ]]
OROR SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
Ex. 3Ex. 3 Find Find SSnn for each of the following:for each of the following:
a) a) aa11 = 58= 58, , aann = -7 = -7, , nn = 26 = 26
SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
SSnn = ½( = ½(2626)[ )[ 5858 - 7- 7 ] ]
Sum of an Arithmetic SeriesSum of an Arithmetic Series
The sum The sum SSnn of the first of the first nn terms of an terms of an
arithmetic series is given by the following:arithmetic series is given by the following:
SSnn = ½ = ½nn[ 2[ 2aa11 + ( + (nn – 1) – 1)d d ]]
OROR SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
Ex. 3Ex. 3 Find Find SSnn for each of the following:for each of the following:
a) a) aa11 = 58, = 58, aann = -7, = -7, nn = 26 = 26
SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
SSnn = ½(26)[ = ½(26)[ 58 - 758 - 7 ] ]
SSnn = ½(26)[ = ½(26)[ 5151 ] ]
Sum of an Arithmetic SeriesSum of an Arithmetic Series
The sum The sum SSnn of the first of the first nn terms of an terms of an arithmetic series is given by the following:arithmetic series is given by the following:
SSnn = ½ = ½nn[ 2[ 2aa11 + ( + (nn – 1) – 1)d d ]]
OROR SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
Ex. 3Ex. 3 Find Find SSnn for each of the following:for each of the following:
a) a) aa11 = 58, = 58, aann = -7, = -7, nn = 26 = 26
SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
SSnn = ½(26)[ 58 - 7 ] = ½(26)[ 58 - 7 ]
SSnn = = ½(26)½(26)[ 51 ][ 51 ]
SSnn = = 1313[ 51 ][ 51 ]
Sum of an Arithmetic SeriesSum of an Arithmetic Series
The sum The sum SSnn of the first of the first nn terms of an terms of an arithmetic series is given by the following:arithmetic series is given by the following:
SSnn = ½ = ½nn[ 2[ 2aa11 + ( + (nn – 1) – 1)d d ]]
OROR SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
Ex. 3Ex. 3 Find Find SSnn for each of the following:for each of the following:
a) a) aa11 = 58, = 58, aann = -7, = -7, nn = 26 = 26
SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
SSnn = ½(26)[ 58 - 7 ] = ½(26)[ 58 - 7 ]
SSnn = ½(26)[ 51 ] = ½(26)[ 51 ]
SSnn = = 13(51)13(51) = = 663663
Ex. 4Ex. 4 1616
∑ ∑ (4(4kk – 2) – 2) kk = 1 = 1
Ex. 4Ex. 4 1616
∑ ∑ (4(4kk – 2) – 2) kk = 1 = 1
nn = 16 = 16
Ex. 4Ex. 4 1616
∑ ∑ (4(4kk – 2) – 2) kk = 1 = 1
nn = 16 = 16
aa11 = 4(1) – 2 = 2= 4(1) – 2 = 2
Ex. 4Ex. 4 1616
∑ ∑ (4(4kk – 2) – 2) kk = 1 = 1
nn = 16 = 16
aa11 = 4(1) – 2 = 2= 4(1) – 2 = 2
aann = 4(16) – 2 = 62 = 4(16) – 2 = 62
Ex. 4Ex. 4 1616
∑ ∑ (4(4kk – 2) – 2) kk = 1 = 1
nn = 16 = 16
aa11 = 4(1) – 2 = 2= 4(1) – 2 = 2
aann = 4(16) – 2 = 62 = 4(16) – 2 = 62
SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
Ex. 4Ex. 4 1616
∑ ∑ (4(4kk – 2) – 2) kk = 1 = 1
nn = 16 = 16
aa11 = 4(1) – 2 = 2= 4(1) – 2 = 2
aann = 4(16) – 2 = 62 = 4(16) – 2 = 62
SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
SSnn = ½( = ½(1616)[)[22 + + 6262]]
Ex. 4Ex. 4 1616
∑ ∑ (4(4kk – 2) – 2) kk = 1 = 1
nn = 16 = 16
aa11 = 4(1) – 2 = 2= 4(1) – 2 = 2
aann = 4(16) – 2 = 62 = 4(16) – 2 = 62
SSnn = ½ = ½nn[ [ aa11 + + aann ] ]
SSnn = ½( = ½(1616)[)[22 + + 6262]]
SSnn = 512 = 512