# Geometric Sequence Problems

### Solutions

1The second term of a geometric sequence is 6, and the fifth term is 48. Determine the sequence.

2The 1st term of a geometric sequence is 3 and the eighth term is 384. Find the common ratio, the sum and the product of the first 8 terms.

3Compute the sum of the first 5 terms of the sequence: 3, 6, 12, 24, 48, ...

4Calculate the sum of the terms of the following geometric sequence:

5Calculate the product of the first 5 terms of the sequence: 3, 6, 12, 24, 48, ...

6John has purchased 20 books. The 1st book cost $1, the 2nd, $2, the 3rd, $4, and the 4th ,$8, and so on. How much did John pay for the 20 books?

7The sides of a square, **l**, have lines drawn between them connecting adjoining sides with their midpoints. This creates another square within the original and this process is continued indefinitely. Calculate the sum of the areas of the infinite squares.

8Calculate the fraction that is equivalent to 0.18181818...

9Calculate the fraction that is equivalent to 3.2777777...

## 1

The second term of a geometric sequence is 6, and the fifth term is 48. Determine the sequence.

a_{2}= 6; a_{5}= 48;

**a _{n} = a_{k} · r^{ n−k}**

48 = 6 r^{5−2} ; r³ = 8; r = 2.

a_{1}= a_{2}/r; a_{1}= 6/2= 3.

3, 6, 12, 24, 48, ...

## 2

The 1st term of a geometric sequence is 3 and the eighth term is 384. Find the common ratio, the sum and the product of the first 8 terms.

a_{ 1 }= 3; a_{ 8 }= 384.

384 = 3 · r^{8−1} ; r^{7} = 128; r^{7} = 2^{7}; r= 2.

S_{8 }= (384 · 2 − 3 ) / (2 − 1) = 765

_{}

## 3

Compute the sum of the first 5 terms of the sequence: 3, 6, 12, 24, 48, ...

## 4

Calculate the sum of the terms of the following geometric sequence:

## 5

Calculate the product of the first 5 terms of the sequence: 3, 6, 12, 24, 48, ...

## 6

John has purchased 20 books. The 1st book cost $1, the 2nd, $2, the 3rd, $4, and the 4th ,$8, and so on. How much did John pay for the 20 books?

a_{1}= 1 r= 2; n = 20;

S= (1 · 2^{20−1} − 1) / (2 − 1) = $1,048,575 .

## 7

The sides of a square, **l**, have lines drawn between them connecting adjoining sides with their midpoints. This creates another square within the original and this process is continued indefinitely. Calculate the sum of the areas of the infinite squares.

## 8

Calculate the fraction that is equivalent to 0.18181818...

0.18181818...= 0.18 + 0.0018 + 0.000018 + ...

a_{1}= 0.18; r= 0.01;

S= 0.18/(1− 0.01)= 2/11

## 9

Calculate the fraction that is equivalent to 3.2777777...

3.2777777...= 3.2 + 0.07 + 0.007 + 0.0007 + ...

a_{1}= 0.07 r= 0.1;

3.2 + 0.07/(1 − 0.1) = 32/10 + 7/90 = 59/18