Partial Sums of Arithmetic & Geometric SequencesSection 14.4

Partial Sums

•Relatively easy to find when n is small.

•Remember:

•What if we want to find the Partial sum of an Arithmetic sequence?

n

i = 1

Partial Sum of an Arithmetic Sequence

The partial sum of the first n terms of an arithmetic sequence is given by

where is the first term of the sequence and is the nth term.

Example 1:Use the partial sum formula to find the sum of the first six terms of the arithmetic sequence 2, 5, 8, 11, 14, 17, …

n = 6, = 2

= 17

OYO:Use the partial sum formula to find the sum of the first five terms of the arithmetic sequence 2, 9, 16, 23, 30.

Example 2:Find the sum of the first 30 positive integers.

Integers: …, -3, -2, -1, 0, 1, 2, 3, …

n = 30, = 1

= 30465

OYO:Find the sum of the first 50 positive integers.

1275

Example 3:Rolls of carpet are stacked in 20 rows with 3 rolls in the top row, 4 rolls in the next row, and so on, forming an arithmetic sequence. Find the total number of carpet rolls if there are 22 rolls in the bottom row.

The list would go: 3, 4, 5, …, 22.

n = 20, = 3

= 22

There are a total of 250 rolls of carpet.

OYO:An ice sculptor is creating a gigantic castle-facade ice sculpture for First Night festivities in Boston. To get the volume of ice necessary, large blocks of ice were stacked atop each other. The topmost row was comprised of 6 blocks of ice, the next row of 7 blocks of ice, and so on, forming an arithmetic sequence. Find the total number of ice blocks needed if there were 15 blocks in the bottom row.

105 blocks of ice

Partial Sum of a Geometric Sequence

The partial sum of the first n terms of an geometric sequence is given by

where is the first term of the sequence, r is the common ratio, and r ≠ 1.

Example 4:Find the sum of the first five terms of the geometric sequence 32, 8, 2,

n = 5, = 32

r =

OYO:Find the sum of the first six terms of the geometric sequence 5, 10, 20, 40, 80, 160.

315

Example 5:A grant from an alumnus to a university specified that the university was to receive $800,000 during the first year and 75% of the preceding year’s donation during each of the following 5 years. Find the total amount donated during the 6 years.

n = 6, = 800,000

r =.75

The total amount donated during the 6 years is $2,630,468.75.

OYO:A new youth center is being established in a downtown urban area. A philanthropic charity has agreed to help it get off the ground. The charity has pledged to donate $250,000 in the first year, with 80% of the preceding year’s donation for each of the following 6 years. Find the total amount donated during the 7 years.

$987,856

Homework

•Unit 19 Homework page #21 - 24