Question

The number of terms in an A.P. is even; the sum of the odd terms in lt is 24 and that the even terms is 30. If the last term exceeds the first term by `10 1/2`, then the number of terms in the A.P. is ______.

पर्याय

• 4

• 8

• 12

• 16

Answer 1

The number of terms in an A.P. is even; the sum of the odd terms in lt is 24 and that the even terms is 30. If the last term exceeds the first term by `10 1/2`, then the number of terms in the A.P. is 8.

Explanation:

Let a, d and 2n be the first term, common difference and total number of terms of an A.P. respectively i.e. a + (a + d) + (a + 2d) + ... + (a + (2n – 1) d)

No. of even terms = n, No. of odd terms = n

Sum of odd terms : S_o = `n/2[2a + (n - 1)(2d)]` = 24

⇒ n [a + (n – 1)d] = 24  ...(i)

Sum of even terms : S_e = `n/2[2(a + d) + (n - 1)2d]` = 30

⇒ n[a + d + (n – 1) d] = 30  ...(ii)

Subtracting equation (i) from (ii), we get

nd = 6  ...(iii)

Also, given that last term exceeds the first term by `21/2`

a + (2n – 1) d = `a + 21/2 2nd - d` = `21/2`

⇒ `2 xx 6 - 21/2` = d  (∵ nd = 6) d = `3/2`

Putting value of d in equation (3) we get n = `(6 xx 2)/3` = 4

∴ Total no. of terms = 2n = 2 × 4 = 8