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Solution


Q.

The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by 212. Then the number of terms which are integers in the A.P. is :          [2025]
1 10  
2 8  
3 4  
4 6  

Ans.

(2)

Let n be the number of terms of an A.P., a be the first term and d be the common difference.

Let n = 2m, so terms are a, a + d, a + 2d, a + (2m – 1)d

Odd terms = a, a + 2d, a + 4d, ...., a + (2m – 2)d

Even terms = a + d, a + 3d, a + 5d, .... a + (2m – 1)d

Now, Seven–Sodd = (a + d – a) + (a + 3d – a – 2d) + .... + (a + (2m – 1)d – a – (2m – 2)d)

⇒ 30–24=md ⇒ md=6          ... (i)

Now, a+(2m–1)d–a=212 ⇒ 2md–d=212

⇒ 12–d=212 ⇒ d=12–212=32          [From (i)]

Since, md=6 ⇒ m×32=6 ⇒ m=4

Now, n=2m=2×4=8

So, number of terms = 8.