Question

Tenth term of Ist A.P. is equal to Twentyth term of 2nd A.P. equal to Thirtyth term of 3rd A.P. Find the ratio of their common difference?

Answer 1

Answer:

let the first term be a and common difference be d

so the terms are a, a+d, a+2d.. a+5d...

2nd term is a+d

3rd term is a+2d

5th term is a+5d

now if these are in GP,

(a+2d) / (a+d) = (a+5d) / (a+2d)

(a+2d)² = (a+d)(a+5d)

a² + 4ad + 4d² = a² + 6ad + 5d²

d² = -2ad (Assuming d is not equal to 0 else we wont have an AP initially)

or d = -2a ... (1)

Now, using this to determine the ratio

(a+2d) / (a+d) = (a-4a) / (a-2a) = -3a/-a = 3

So,

common ratio is 3.

Answer 2

Step-by-step explanation:

$<b>$

so the terms are a, a+d, a+2d.. a+5d...

2nd term is a+d

3rd term is a+2d

5th term is a+5d

now if these are in GP,

$\sf\frac{(a+2d)}{(a+d)} = \frac{(a+5d)}{(a+2d)}$

(a+2d)² = (a+d)(a+5d)

a² + 4ad + 4d² = a² + 6ad + 5d²

d² = -2ad (Assuming d is not equal to 0 else we wont have an AP initially)

or d = -2a ... (1)

Now, using this to determine the ratio

(a+2d) / (a+d) = (a-4a) / (a-2a) = -3a/-a = 3

So,

common ratio is 3.