Question

The 25th term of an arithmetic series is 44.5

The sum of the first 30 terms of this arithmetic series is 765

Find the 16th term of the arithmetic series.

Show your working clearly.

Answer 1

`a_(16)=26.5`

Explanation:

Arithmetic series:
`a_n=a+(n-1)d`

Sum of arithmetic series:
`S_n=(n)/(2)[2a+(n-1)d]`

(where a is the first term and d is the common difference)

Given:

• 
  `a_(25)=44.5`
• 
  `S_(30)=765`

`\implies a_(25)=44.5`

`\implies a+(25-1)d=44.5`

`\implies a+24d=44.5`

`\implies S_(30)=765`

`\implies (30)/(2)[2a+(30-1)d]=765`

`\implies 30a+435d=765`

Rewrite
`a+24d=44.5` to make a the subject:

`\implies a=44.5-24d`

Substitute found expression for a into
`30a+435d=765` and solve for d:

`\implies 30(44.5-24d)+435d=765`

`\implies 1335-285d=765`

`\implies 285d=570`

`\implies d=2`

Substitute found value of d into
`a=44.5-24d` and solve for a:

`\implies a=44.5-24(2)=-3.5`

Therefore:

`\implies a_(16)=-3.5+(16-1)2=26.5`