Question

The 5th term of an arithmetic series is 13 and the 15th term is 33. Find and simplify an

expression for the sum of n terms.

Answer 1

aₙ = 2n+ 3

Explanation:

The general equation for the nth term of an arithmetic series with common difference d and first term a(1) is given by

aₙ = a₁ + d(n - 1)

Using values for the 5th and 15th terms we get

a₅ = a₁ + d(5 -1) = 13
==> a₁ + 4d = 13 (1)

a₁₅ = a₁ + d(15 - 1) = 33
a₁ + 14d = 33 (2)

Eq (2) - Eq(1) gives us:
a₁ + 14d - (a₁ + 4d) = 33 - 13

a₁ + 14d - a₁ - 4d = 20

a₁ - a₁ + 14d - 4d =20

10d = 20

d = 2

Plugging d = 2 in (1) we get:
a₁ + 4(2) = 13
a₁ + 8 = 13

a₁ = 13 - 8 =5

So first term is 5,common difference is 2

Expression for nth term:
aₙ = 5 + 2(n - 1)

aₙ = 5 + 2n - 2

aₙ = 2n+ 3