Question

Complete the pattern: 2,5,10,17,___,____,____ and find Nth term in the sequence.

Answer 1

The differences between the successive terms of the given series are 3,5,7,…Apparently, these differences are in arithmetic progression.

If the initial term of such a series is a₀ and if d_n is the difference between the nth term and the (n+1)th term , then a_n = a₀+(n-1)d_n and d_n= d₀+(n-2)d where d₀ is the difference between the 1^st and the 2^nd terms and d is the common difference in the arithmetic progression formed by the differences between the of the successive terms of the series {a_n}. Thus, a_n=a₀+(n-1)[d₀+(n-2)d] = a0+(n?1)d0+ (n?1)(n?2)d.

In case of the given series, d₀ = 3 and d = 2 so that a_n = 2+3(n-1)+2(n-1)(n-2) = (3n-1)+ 2(n-1)(n-2). The series is 2,5,10,17,26,37,50,65,82,101,….