INFINITE GEOMETRIC SEQUENCE HELP
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INFINITE GEOMETRIC SEQUENCE HELP

A) determine whether the given sequence is geometric. If it is geometric find the common ratio.

{(5/4))^n}

B) Find the 7th term of the following geometric sequence

0.1,1.0,10.0,.........

C) Find the sum of the following infinite geometric sequence.

1-3/4+9/16-27/64+..................

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INFINITE GEOMETRIC SEQUENCE HELP

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Apr 21, 2011
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starstarstarstarstar 		Infinite Geometric Sequence
by: s

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Part II

C) Find the sum of the following infinite geometric sequence.

1-3/4+9/16-27/64+..................

a_n = 1, -3/4, 9/16, -27/64, . . .


r, the common ration is:

r_n = a_3/a_2
= -27/64 ÷ 9/16 = -432/576 = -.75


the geometric sequence is:

a_n = a_1 * (r)^(n - 1)

a_n = 1 * (-.75)^(n - 1)


this infinite geometric series is:

S = 1 - ¾ + 9/16 - 27/64 + …

an infinite geometric series will converge if:

-1 < r < +1

r = -0.75; this series will converge

(this series will converge because each consecutive term gets smaller and smaller)


the general formula for the sum of an infinite geometric series is:

S_∞ = a_1/(1 - r)


The sum of this infinite geometric series is:

S_∞ = 1/[1 - (-.75)]

S_∞ = 1/(1 + .75)

S_∞ = 1/(1.75)

S_∞ = 0.571429

The final answer is: S_∞ = 0.571429



Thanks for writing.


Staff
www.solving-math-problems.com


Apr 21, 2011
Rating
starstarstarstarstar 		Infinite Geometric Sequence
by: Staff


Part I

The question:

A) determine whether the given sequence is geometric. If it is geometric find the common ratio.

{(5/4))^n}

B) Find the 7th term of the following geometric sequence

0.1,1.0,10.0,.........

C) Find the sum of the following infinite geometric sequence.

1-3/4+9/16-27/64+..................


The answer:

A) determine whether the given sequence is geometric. If it is geometric find the common ratio.

{(5/4))^n}


There is an extra right parenthesis in {(5/4))^n}. I think this is what you mean:


a_n = (5/4)^n


the first four terms in the sequence are:


if n = 1, the first term is

a_1 = (5/4)¹

a_1 = 5/4


if n = 2, the second term is

a_2 = (5/4)²

a_2 = 5²/4²

a_2 = 25/16


if n = 3, the third term is

a_3 = (5/4)³

a_3 = 5³/4³

a_3 = 125/64


if n = 4, the fourth term is

a_4 = (5/4)⁴

a_4 = 5⁴/4⁴

a_4 = 625/256


a_n = 5/4, 25/16, 125/64, 625/256, . . .

“If” this is a geometric sequence, then r, the common ratio, will be the same no matter what two consecutive terms are used to calculate the ratio:

r_n = a_n/a_n-1

(n must be greater than 1)

r_n = r with a subscript of n (this is the common ratio)

a_n = a with a subscript of n (this is the nth term in the sequence)

a_n-1 = a with a subscript of n-1 (this is the n-1 term in the sequence)

using the 1st and 2nd terms in the sequence to calculate the common ratio

r_n = a_2/a_1

25/16 ÷ 5/4 = 100/80 = 5/4 = 1.25

using the 2nd and 3rd terms in the sequence to calculate the common ratio

r_n = a_3/a_2

125/64 ÷ 25/16= 2000/1600 = 5/4 = 1.25

using the 3rd and 4th terms in the sequence to calculate the common ratio

r_n = a_4/a_3

625/256 ÷ 125/64 = 40000/32000 = 5/4 = 1.25


The calculated ratios all have the same value:

r_n = a_2/a_1 = a_3/a_2 = a_4/a_3 = 5/4 = 1.25

Therefore, the sequence a_n = (5/4)ⁿ is a geometric sequence


B) Find the 7th term of the following geometric sequence

0.1,1.0,10.0,.........

using the 2nd and 3rd terms in the sequence to calculate the common ratio

r_n = a_3/a_2

10.0 ÷ 1.0 = 10

The common ratio, r, is equal to 10.


A geometric sequence has the (general) form:

a_n = a_1 * (r)^(n - 1)


a_n = a with a subscript of n (this is the nth term in the sequence)

a_1 = a with a subscript of 1 (this is the 1st term in the sequence)

n = number of terms

r = the common ratio

The equation for this geometric sequence is:

a_1 = 0.1

r = 10

a_n = a_1 * (r)^(n - 1)

a_n = 0.1 * (10)^(n - 1)


The 7th term in the sequence is:

a_7 = 0.1 * (10)^(7 - 1)

a_7 = 0.1 * (10)⁶

a_7 = (10)⁵

a_7 = 100,000

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