Question

the 8th term of an arithmetic sequence is -42 and the 16th term is -74. Find and simplify an expression for the nth term

Answer 1

`a_(n)` = - 4n - 10

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

given a₈ = - 42 and a₁₆ = - 74 , then

a₁ + 7d = - 42 → (1)

a₁ + 15d = - 74 → (2)

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

8d = - 74 - (- 42) = - 74 + 42 = - 32 ( divide both sides by 8 )

d = - 4

substitute d = - 4 into (1) and solve for a₁

a₁ + 7(- 4) = - 42

a₁ - 28 = - 42 ( add 28 to both sides )

a₁ = - 14

Then

`a_(n)` = - 14 - 4(n - 1) = - 14 - 4n + 4 = - 4n - 10

Answer 2

`a_(n)=-10-4n`

Step-by-step explanation

`a_(n)=a_(1)+(n-1)d\\Given:a_(8)=-42\\&a_(16)=-74\\Hence: \\a_(8)=a_(1)+7d=-42 (1)a_(16)=a_(1)+15d=-74 (2)\\ \left \{ {{a_(1)+7d=-42} \atop a_(1)+15=-74}} \right. \\(2)-(1):8d=-32\\ d=-32/8 =-4\\Substitute :d=-4 in equation (1)`