For the sequence given by a n = 4 n 2 - n + 1 , find first three terms.

Given the sequence a n = 4 n 2 - n + 1 Let the first three terms are denoted as a 1 , a 2 , a 3 respectively. So, a 1 = 4 1 2 - 1 + 1 = 4 a 2 = 4 2 2 - 2 + 1 = 16 - 1 = 15 a 3 = 4 3 2 - 3 + 1 = 36 - 2 = 34 .

If 2 3 , k , 5 8 k are in A . P . , find the value of k .

Given that 2 3 , k , 5 8 k are in A . P . ⇒ 5 8 k - k = k - 2 3 ⇒ 2 3 = k + k - 5 8 k ⇒ 2 3 = 16 k - 5 k 8 = 11 k 8 ⇒ k = 2 3 × 8 11 = 16 33 .

Find the sum of the series 1 - a + a 2 - … to n terms ( a ≠ - 1 ) .

Given the series : 1 - a + a 2 - … to n terms ( a ≠ - 1 ) . Since , sum of a G . P . series is represented by the formula, S n = a 1 r n - 1 r - 1 , when r ≠ 1 . S n represents the sum of the G . P . series upto n t h terms, a 1 represents the first term, r represents the common ratio and n represents the number of terms. Here, a 1 = 1 and r = - a 1 = - a ∴ S n = 1 × ( - a ) n - 1 - a - 1 ⇒ S n = 1 - ( - a ) n 1 + a

Find the sum of the sequence - 5 4 , 5 16 , - 5 64 , …

Given the sequence - 5 4 , 5 16 , - 5 64 , … We have to find out S = - 5 4 + 5 16 - 5 64 + . . . . . , which is an infinite G . P . with first term a = - 5 4 and common ratio r = 5 / 16 - 5 / 4 = - 1 4 Since , S = a 1 - r = - 5 4 1 - - 1 4 = - 5 4 × 4 5 = - 1 .