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in Arithmetic Progression by (22.8k points)
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If the sum of 2n terms of the A.P. 2, 5, 8, 11,... is equal to the sum of n terms of A.P. 57, 59, 61, 63, ..., then n is equal to

(a) 10 

(b) 11 

(c) 12 

(d) 13

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Answer : (b) 11

For the 1st A.P., 2, 5, 8, 11, ...... , 

First term (a1) = 2, common difference (d1) = 3 

∴ The sum of this A.P. to 2n terms

\( \frac{2n}{2}[2a_1+(2n-1)d_1]\)

= n[4+(2n - 1)3]

= 4n + 6n2 – 3n = 6n2 + n = n (6n + 1)

For the second A.P., 57, 59, 61, 63, ....... , 

First term (b1) = 57, common difference (d2) = 2 

∴ The sum of this A.P. to n term is

\( Sum_n = \frac{n}{2}(2b_1+(n-1)d_2]\)

\(\frac{n}{2}\)[114 + (n - 1) 2]

= 57n + n2 – n = n2 + 56n = n (n + 56)

Given, S2n = Sumn

⇒ n(6n + 1) = n(n + 56) 

⇒ 6n + 1 = n + 56 

⇒ 5n = 55 

n = 11.

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