Question

Use the formula for sum of first n terms of a geometric sequence to evaluate following sum. (Hint: Find the first term a and common ratio r for each of the given sum.)

a) 10:i=1 Σ (5^i)/(3^(i-1))

b) 15:i=1 Σ (2^(2i))/(3^(i+1))

(a) Find sum of first 7 terms of the geometric sequence whose 5th term is 32 and 6th term is 64.

(b) Find sum of first 5 terms of the geometric sequence whose 4th term is 9/4 and 5th term is 27/8 .

(c) Find sum of first 6 terms of the geometric sequence whose 2 nd term is 3 and 5 th term is 81.

Answer 1

(a)

Step 1:

The sequence is .

The terms of the sequence are:

Substitute .

.

Substitute .

.

Substitute .

.

Therefore, the geometric sequence is .

Step 2:

The common ratio is .

.

The first term of the sequence is .

Sum of first terms in a geometric series is .

Substitute , and in .

The sum of the series is .

Solution:

The sum of the series is .

Answer 2

(b)

Step 1:

The sequence is .

The terms of the sequence are:

Substitute .

.

Substitute .

.

Substitute .

.

Therefore, the geometric sequence is .

Step 2:

Find the sum of first terms of the sequence.

The common ratio is .

.

The first term of the sequence is .

Sum of first terms in a geometric series is .

Substitute , and in .

The sum of the sequence is .

Solution:

The sum of the sequence is .

Answer 3

(2a)

Step 1:

The term of the geometric sequence is .

The term of the geometric sequence is .

The common ratio is .

Formula for term in geometric sequence is .

Find the first term of the geometric sequence .

Substitute and in .

.

Substitute .

.

The first term of the sequence is .

Step 2:

Find the sum of first terms.

The sum of first terms in a geometric series is .

Substitute , and in .

.

The sum of first terms is .

Solution:

The sum of first terms is .

Answer 4

(2b)

Step 1:

The term of the geometric sequence is .

The term of the geometric sequence is .

The common ratio is .

Formula for term in geometric sequence is .

Find the first term of the geometric sequence .

Substitute and in .

Substitute .

.

The first term of the sequence is .

Step 2:

Find the sum of first terms.

The sum of first terms in a geometric series is .

Substitute , and in .

.

The sum of first terms is .

Solution:

The sum of first terms is .

Answer 5

(2c)

Step 1:

The term of the geometric sequence is .

The term of the geometric sequence is .

Formula for term in geometric sequence is .

Find the first term and common ratio of the geometric sequence.

Substitute in .

Substitute .

.

Substitute in .

Substitute .

Substitute .

The common ratio is .

Substitute in .

.

The first term of the sequence is .

Step 2:

Find the sum of first terms.

The sum of first terms in a geometric series is .

Substitute , and in .

.

The sum of first terms is .

Solution:

The sum of first terms is .