Question

How to find first term and commond difference if the only given are the last term (80) and sum of first 10 terms (530)?

Answer 1

`color(blue)("First term"=26)`

`color(blue)("Common difference"=6)`

Explanation:

This is an arithmetic series.

The `nth` term of an arithmetic series is given by:

`a+(n-1)d \ \ [1]`

Where `bba` is the first term, `bbd` is the common difference and `bbn` is the nth term.

The sum of an arithmetic series is given as:

`S_n=n/2(2a+(n-1)d) \ \ \ [2]`

We are given:

Sum of the first 10 terms is `530` and `n=10`

Using in `[2]`

`530=10/2(2a+(10-1)d)`

`530=10a+45d \ \ \ [3]`

Last term is 80:

Using this in `[1]`

`a+(10-1)d=80 \ \ \ [2]`

`a+9d=80 \ \ \ [4]`

Solving `[3]` and `[4]` simultaneously:

`a+9d=80=>a=80-9d`

in `[3]`

`530=10(80-9d)+45d`

`530-800=-45d=>d=6`

Substituting in `[4]`

`a+9(6)=80`

`a=80-54=26`

First term is:

`color(blue)(26)`

The common difference is:

`color(blue)(60`

Check:

last term:

`26+(10-1)(6)=80`

Sum of first 10 terms:

`10/2(2(26)+(10-1)(6))=10/2(52+54)=5(106)=530`