Question

What is the recursive formula for this geometric sequence 2, -10, 50, -250

Answer 1

`T_n = \left \{ {{T_1=2} \atop {T_n =-5T_(n-1)}} n>1 \right.`

Explanation:

Given

Geometric sequence: 2, -10, 50, -250

Required

Recursive formula of the sequence

We start by naming each sequence

`T_1 = 2\\T_2 = -10\\T_3 = 50\\T_4 = -250`

Represent each term in relation to the previous term (except T1)

`T_1 = 2\\T_2 =2 * -5\\T_3 = -10 * -5\\T_4 = 50 * -5`

Recall that

`T_1 = 2\\T_2 = -10\\T_3 = 50\\T_4 = -250`

So, at this point; we have to make substitutions

`T_1 = 2\\T_2 =T_1 * -5\\T_3 = T_2 * -5\\T_4 = T_3 * -5`

Subtract 1 from each term on the right hand side

`T_1 = 2\\T_2 =T_(2-1) * -5\\T_3 = T_(3-1) * -5\\T_4 = T_(4-1) * -5`

Replace each term greater than 1 by n

`T_1 = 2\\T_n =T_(n-1) * -5\\T_n = T_(n-1) * -5\\T_n = T_(n-1) * -5`

Remove repetition

`T_1 = 2\\T_n =T_(n-1) * -5`

`T_1 = 2\\T_n =-5T_(n-1)`

Hence, the recursive formula is

`T_n = \left \{ {{T_1=2} \atop {T_n =-5T_(n-1)}} n>1 \right.`