A is an arithmetic progression of 13 terms. G is a geometric progression of 13 terms. The product of the terms of G is 8192. The sum of the terms of A is 26. Find the sum of the seventh terms of A and G.a) 6b) 4c) 10d) Cannot be determinedCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

Given:
- An arithmetic progression A of 13 terms
- A geometric progression G of 13 terms with product 8192
- Sum of terms of A is 26

To find:
- Sum of seventh terms of A and G

Solution:
1. Let the first term of A be a and the common difference be d. Then, the sum of terms of A is given by:
S(A) = (n/2)(2a + (n-1)d)
where n = 13 (number of terms)
S(A) = (13/2)(2a + 12d) = 26
Simplifying, we get: 2a + 12d = 4

2. Let the first term of G be b and the common ratio be r. Then, the product of terms of G is given by:
P(G) = b^n * r^(n(n-1)/2)
where n = 13 (number of terms)
P(G) = b^13 * r^78 = 8192
Simplifying, we get: br^6 = 2

3. We need to find the sum of seventh terms of A and G.
- The seventh term of A is given by: a + 6d
- The seventh term of G is given by: b*r^6

4. To solve for a, d, b, and r, we can use simultaneous equations from step 1 and 2:
2a + 12d = 4
br^6 = 2
Dividing the first equation by 2, we get: a + 6d = 2
Substituting br^6 = 2, we get: a = 2/r^6 - 6d

5. Substituting the value of a in terms of d and r in the first equation, we get:
2(2/r^6 - 6d) + 12d = 4
Simplifying, we get: 4/r^6 + 6d = 2
Dividing by 2, we get: 2/r^6 + 3d = 1

6. Substituting the value of d in terms of r from step 2, we get:
2/r^6 + 3(2/br^6) = 1
Simplifying, we get: r^6 + 6r^6 = 1
Solving for r, we get: r = 1/2 or r = -1/2

7. Since r cannot be negative (as it is a term in a geometric progression), we get: r = 1/2
Substituting r in br^6 = 2, we get: b = 4

8. Substituting a, d, b, and r in the formulas for the seventh terms of A and G, we get:
Seventh term of A: a + 6d = 2/r^6 - 6d + 6d = 2/r^6 = 128
Seventh term of G: b*r^6 = 4*1/2^6 = 1/4

9. The sum of seventh terms of A and

Similar Electronics and Communication Engineering (ECE) Doubts

The sum of the squares of three terms in arithmetic progression is 365. The product of the first and the third terms is 120. Find the sum of the squares of the second term and the common difference.

There are 30 terms in an arithmetic progression. The second and third terms are distinct integers. The ratio of the sum of first 20 terms and the sum of the first 10 terms equals twice the ratio of the second and first terms. Which of the following can be the sum of all its terms?

The number of product terms in the minimised sum-of-product expression obtained through the following K-map is(where "d" denotes do not care states)(Answer up to the nearest integer) Correct answer is '2'. Can you explain this answer?

When two time constant elements are cascaded non-interactively then, the overall transfer function of such an arrangement

If the arithmetic mean of two numbers is 10 and their geometric mean is 8, the numbers are

Top Courses for Electronics and Communication Engineering (ECE)View all

GATE ECE (Electronics) Mock Test Series 2025

Network Theory (Electric Circuits)

General Aptitude for GATE

Crash Course: Electronic Communication Engineering (ECE)

Signals and Systems

Top Courses for Electronics and Communication Engineering (ECE)

Top Courses for Electronics and Communication Engineering (ECE)

Suggested Free Tests

Related Content

Mock Test Series

Subject-wise Courses

Subject-wise Courses

Other Engg Branches

Company