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Use the formula for sum of first n terms of a geometric sequence to evaluate following sum.

0 votes

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.

asked Nov 29, 2015 in CALCULUS by anonymous

5 Answers

0 votes

(a)

Step 1:

The sequence is image.

The terms of the sequence are:

Substitute image.

image.

Substitute image.

image.

Substitute image.

image.

Therefore, the geometric sequence is image.

Step 2:

The common ratio is image.

image.

The first term of the sequence is image.

Sum of first image terms in a geometric series is image.

Substitute image, image and image in image.

image

image

image

image

image

The sum of the series is image.

Solution:

The sum of the series is image.

answered Nov 30, 2015 by Sammi Mentor
edited Nov 30, 2015 by Sammi
0 votes

(b)

Step 1:

The sequence is image.

The terms of the sequence are:

Substitute image.

image.

Substitute image.

image.

Substitute image.

image.

Therefore, the geometric sequence is image.

Step 2:

Find the sum of first image terms of the sequence.

The common ratio is image.

image.

The first term of the sequence is image.

Sum of first image terms in a geometric series is image.

Substitute image, image and image in image.

image

image

image

image

image

image

The sum of the sequence is image.

Solution:

The sum of the sequence is image.

answered Nov 30, 2015 by Sammi Mentor
edited Nov 30, 2015 by Sammi
0 votes

(2a)

Step 1:

The image term of the geometric sequence is image.

The image term of the geometric sequence is image.

The common ratio is image.

Formula for term in geometric sequence is .

Find the first term of the geometric sequence .

Substitute image and image in .

image.

Substitute image.

image

image

image.

The first term of the sequence is image.

Step 2:

Find the sum of first image terms.

The sum of first terms in a geometric series is .

Substitute image, image and image in .

image

image

image

image

image.

The sum of first image terms is image.

Solution:

The sum of first image terms is image.

answered Nov 30, 2015 by Sammi Mentor
0 votes

(2b)

Step 1:

The image term of the geometric sequence is image.

The image term of the geometric sequence is image.

The common ratio is image.

Formula for term in geometric sequence is .

Find the first term of the geometric sequence .

Substitute image and image in .

image

Substitute image.

image

image

image

image.

The first term of the sequence is image.

Step 2:

Find the sum of first image terms.

The sum of first terms in a geometric series is .

Substitute image, image and image in .

image

image

image

image.

The sum of first image terms is image.

Solution:

The sum of first image terms is image.

answered Nov 30, 2015 by Sammi Mentor
edited Nov 30, 2015 by Sammi
0 votes

(2c)

Step 1:

The image term of the geometric sequence is image.

The term of the geometric sequence is image.

Formula for term in geometric sequence is .

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

Substitute image in .

image

Substitute image.

image

image.

Substitute image in .

image

Substitute image.

image

image

Substitute image.

image

image

image

image

The common ratio is image.

Substitute image in image.

image.

The first term of the sequence is image.

Step 2:

Find the sum of first image terms.

The sum of first terms in a geometric series is .

Substitute image, image and image in .

image

image

image

image.

The sum of first image terms is image.

Solution:

The sum of first image terms is image.

answered Nov 30, 2015 by Sammi Mentor
edited Nov 30, 2015 by Sammi

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