Geometric Sequence Sum

A function defined from the set of natural numbers to the set of real numbers is called sequence. Let "a" denote the sequence. Let "n" be a natural number. The image of n under the sequence a is given by a(n) = an.Domain of any sequence is the set of natural numbers. The images of 1, 2, 3,... n ... of the sequence "a" are denoted by a1, a2, a3,... an, ... respectively. So, the sequence is a1, a2, a3,... an, ...
Introduction to geometric sequence:

A sequence whose succeeding term is obtained by multiplying a constant number (except zero)  to the preceding term is called as geometric sequence.

Ex 1: 1, 3, 9, 27, ...

The second term  "3" is obtained by multiplying 3 with the preceding term 1.

The third term “9” is obtained by multiplying 3 with the second term 3.

The fourth term “27” is obtained by multiplying 3 with the third term 9.

And so on…


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Ex 2:   `1/2, 1/6, 1/18,1/54...`

The second term  "1/6" is obtained by multiplying 1/3 (or dividing by 3) with the preceding term 1/2.

The third term “1/18” is obtained by multiplying 1/3 with the second term 1/6.

The fourth term “1/54” is obtained by multiplying 1/3 with the third term 1/18.

And so on…

The following image explains the same concept.

geometric sequence
Examples Involving the Terms of Geometric Sum and Sequence:

Ex 1: Find the three numbers in geometric sequence whose sum is 14 and product is 64.

Sol:

Step 1: Assign the three terms of the geometric sequence.

Let the numbers be `a/r,a,ar`

Step 2: Product is given as 64

`a/rxxaxxar =64`

a3 = 64

`=> a = 4`

Step 3: Sum of the three numbers is 14

`a/r+a+ar =14`

`a(1/r+1+r)=14`

Plug a =4

`4((1+r+r^2)/r)=14`

`rArr`  2r2 - 5r + 2 = 0

`:.` r = 1/2 or 2

Step 4: Find the three numbers.

If r =2, then the numbers are 2, 4, 8.

If r = ½, the numbers are 8, 4, 2.

Ex 2: Find three numbers in geometric sequence such that their sum is 7 and the sum of  their reciprocals is 7/4.

Sol:

Step 1: Let the three numbers be a, ar, ar2.

Step 2: Sum of the numbers = a + ar + ar2

Given: a + ar + ar2 =7

a(1 + r + r2) = 7       ---  (1)

Step 3: Sum of their reciprocals = `1/a+1/(ar)+1/(ar^2) = 7/4=(1+r+r^2)/(ar^2)`        --- (2)

Step 4: Divide (1) and (2) we get,

(ar)2 = 4, ar = ± 2 => a = `+-2/r`

Step 5: Plug a = 2/r in (1)

`2/r(1+r+r^2)=7=>2(1+r+r^2)=7r`

`rArr` 2r2 - 5r + 2 = 0

`rArr` r = 1/2 or 2

Step 6: If r = ½ then a = 4

The numbers are 4, 2, 1, ...

Step 7: If r = 2 then a = 1

The numbers are 1, 2, 4 ...

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Problems on Sum in Geometric Sequence

The sum of the three numbers of a geometric sequence. is 26. Their product is 216. Find the numbers.

Sol: 18, 6, 2 or 2, 6, 18

Find three numbers in geometric sequence such that their sum is 19/3 and their sum of their reciprocals is 19/12.

Sol: 3, 2, 4/3