Arithmetic and Geometric Series

13-3 Arithmetic and Geometric Series and Their Sums

Try the quiz at the bottom of the page! go to quiz

Let's "walk" on thru some series and sums!!

Important differences:

Finite sequence: 1, 5, 9, 13, 17 Finite series: 1 + 5 + 9 + 13 Infinite sequence: 1, 2, 4, 8, 16, . . . Infinite series: 1 + 2 + 4 + 8 + 16 + ^{. . .}

Note that a series is an indicated sum of the terms of a sequence!! In this section, we work only with finite series and the related sums.

How to find the sum of a finite Arithmetic Series!

s_n = n(t₁ + t_n)/2 To find the sum of a finite arithmetic series, you need to know three things. The first term, the last term and the number of terms. Example problem:

1) Find the sum of the first 30 terms of 5 + 9 + 13 + 17 + ^{. . .}

Answer: n = 30, that's pretty obvious!

t₁ = 5, and that's pretty obvious!

We need the 30th term. Use the defintion of an arithmetic sequence.

t₃₀ = 5 + 29^.4 = 121

Therefore: S₃₀ = 30(5 + 121)/2 = 1890

The count says, "I love these problems cause they make me count!"

How to find the sum of a finite Geometric Series

S_n = t₁(1 - rⁿ)/(1 - r) where r is the common ratio and (r doesn't = 0)

To find the sum of a finite geometric series, you need to know three things: the first term, how many terms to add and the common ratio!! (piece of cake!) Example problem:

2) Find the sum of the first 10 terms of the geometric series:
4, 8, 16, 32, 64, ^{. .
.}

Answer: t₁ = 4

r = 2

t₁₀ = 4 ^. 2⁹ = 2048 (This is the formula for a geometric sequence!)

Therefore: S₁₀ = 4(1-2¹⁰)/(1 - 2) = 4 ^. 1023 = 4092

The count restates, " Counting is such great fun! I hate to see this come to an end! But Mr. K says we will be counting infinitely soon!! I can't wait! I will start counting now! 1 advanced math student, 2 advanced math students, three advanced math students, . . ."

On to the next section! We will begin out study of limits in the next section as related to infinite sequences. I think we must be getting close to some calculus. What do you think? See you in the next section!

Current quizaroo # 13a

a) not arithmetic; b) t_n = 4n - 1; c) t_n = 3n + 1; d) t_n = t_n-1 + n; e) t_n = 3n - 1

: b) t_n = n² + 1; c) t_n = 2n + 1; d) t_n = t_n-1 + 2n + 3; e) t_n = n² - 1
3) Find the recursive formula for the sequence: 3, 13, 33, 73, 153, . . .
: a) t₁ = 3, t_n = t_n-1 + 10n; b) t_n = 2t_n-1 + 7; c) t₁ = 3, t_n = 2t_n-1 + 7; d) t_n = 3 + 10n; e) There is no formula


4) Find the sum of the arithmetic series: S₅₀; 6 + 12 + 18 + 24 + 30 + ^{. . .}
: a) 15300; b) 306; c) 153; d) 7500; e) 7650

5) Find the sum of all the multiples of 4 between 1 and 999.

a) 249

b) 249000

c) 124500

d) 1000

e) 124002

click here for answers!!