Question

Identify the 42nd term of an arithmetic sequence where a_1=-12 and a_27=66

Answer 1

111

Explanation:

Arithmetic sequences are linear. That means no matter the pair of points use to calculate slope, or common difference, that number remains constant.

We are given points (1,-12) and (27,66) are on the line.

Calculating slope by subtracting points and then putting 2nd number over 1st.

(27,66)

(1, -12)

-----------subtracting

26 , 78

The slooe or the common difference is 78/26=3.

So now we want to know n such that (42,n) is on this same line.

Let's use the slope formula again with (42,n) and (1,-12).

(42,n)

( 1,-12)

----------subtracting

41 , n+12

The slope is (n+12)/41 but we have also calculated it to be 3 so these expressions are equal.

(n+12)/41=3

Multiply both sides by 41:

n+12=123

Subtract 12 on both sides.

n=111

Answer 2

a₄₂ = 107

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 a₂₇ = 66 , then

a₁ + 26d = 66 , that is

- 12 + 26d = 66 ( add 12 to both sides )

26d = 78 ( divide both sides by 26 )

d = 3

Then

a₄₂ = - 16 + (41 × 3) = - 12 + 123 = 111