Question

Find the sum of 5-7+8-9+11-11+14-13 up to 20 terms?

Answer 1

`25`

Explanation:

Given the series:

`5-7+8-9+11-11+14-13+...`

Note that the odd terms are:

`5, 8, 11, 14,...`

which is an arithmetic sequence with initial term `5` and common difference `3`.

The `10`th term of this subsequence is `5+3(10-1) = 5+27 = 32` and the sum to `10` terms is hence:

`10 * (5+32)/2 = 185`

Meanwhile, the even terms are:

`-7, -9, -11, -13,...`

which is an arithmetic sequence with initial term `-7` and common difference `-2`.

The `10`th term of this subsequence is `-7-2(10-1) = -25` and the sum to `10` terms is hence:

`10 * (-7+(-25))/2 = -160`

So the sum of both subsequences is `185-160 = 25`

Alternatively (and as a check), note the result of combining pairs of terms of the original sequence:

`5-7+8-9+11-11+14-13+... = (-2)+(-1)+0+1+...`

which is an arithmetic series with initial term `-2` and common difference `1`.

The `10`th term of this series is `-2+1(10-1) = 7` and the sum to `10` terms is hence:

`10 * ((-2)+7)/2 = 25`