Question

If the sum of the first 2n terms of the ap series 2, 5, 8,....., is equal to the sum of the first n terms of the ap series 57, 59, 61,...., then n equals (iitjeeadvanced, 2001)

Answer 1

2,5,8,..... is an AP with first term a = 2 and common diference d = 3  

Sum(2n) = (2n/2) ( 2a +(2n-1)(d) )  

= n ( 2a + (2n-1) d)  

= n (4+(2n-1)(3))  

= 4n+ 6n^2 -3n  

= 6n^2 + n -----(1)  

57,59,61,... is an AP with a=57 and d = 2  

sum(n) = (n/2) ( 2a +(n-1) d )  

= (n/2) ( 2(57) + (n-1)(2) )  

= (n/2) (114 + 2n-2)  

= (n/2) (112+2n)  

= 56n +n^2 ------(2)  

(1) = (2)  

6n^2+n = 56n+n^2  

5n^2-55n = 0  

5n(n-11) =0  

n=11