Find the 4th term if the sequend

Question

Find the 4th term if the sequend in which a1 = 2 and an+1 = -4an + 2.

Answer 1

The sequence begins with a₁=2.   The terms are described as a_n, where n is the n^th tern.  That is, a₁ is the first term and a₄ is the fourth term.

 

To find the second term, the first term is used in the formula  a_n+1 = -4a_n + 2.  So, a₂  = -4(2) + 2 = -6.

Third term:   a₃ = -4(-6) + 2 = 26

Fourth term:  a₄ = -4(26) + 2 = -102

 

 

Special note:  Many mathematicians, engineers, computer scientists, etc. use a₀ as the first term in a sequence, the first element to be summed, the first element in an array, etc -- others use 1 as the index of the first element.  This problem gives a₁ as a constant (and not with the recursive formula definition), so the author of this problem is starting with an index of 1.

Answer 2

The general formula is: a_n+1=-4a_n+2. Each term can be found by assigning the corresponding value to 'n" as follows:

n=0 --> a₁=-4a0+2 as a given a1 = 2 so a0=0 (this is your first term and a1 is your 2nd term).

n=1 --> a2 =-4(a1)+2, a1 = 2 then a2 = -4×2+2 = -6 (3rd term)

n=2 --> a3 =-4(a2)+2, a2 = -6 then a3 = -4×-6+2 = 26 (4th term)