Question

the third and fifth terms of an arithmetic sequence are 2 and 32, respectively. find the explicit and recursive formulas for the sequence

Answer 1

see explanation

Explanation:

the nth term of an arithmetic sequence is

`a_(n)` = a₁ + (n - 1)d

where a₁ is the first term and d the common difference

We require to find a₁ and d

given a₃ = 2 and a₅ = 32 , then

a₁ + 4d = 32 → (1)

a₁ + 2d = 2 → (2)

subtract (2) from (1) term by term to eliminate a₁

0 + 2d = 30

2d = 30 ( divide both sides by 2 )

d = 15

substitute d = 15 into (2) and solve for a₁

a₁ + 2(15) = 2

a₁ + 30 = 2 ( subtract 30 from both sides )

a₁ = - 28

Then

`a_(n)` = - 28 + 15(n - 1) = - 28 + 15n - 15 = 15n - 43

Explicit formula is
`a_(n)` = 15n - 43

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A recursive formula allows a term in the sequence to be found by adding d to the previous term, that is

`a_(n)` =
`a_(n-1)` + 15 ; a₁ = - 28