Question

Express

`14x + (12x + y) + (10x + 2y) + ... + 7y`
In Σ notation.

Answer 1

The coefficient on
`x` starts at 14 and decreases by 2 until it reaches 0. Similarly, the coefficient on
`y` starts at 0 and increases by 1 until it reaches 7.

So the general
`n`-th term of the sum might by

`(14-2(n-1))x+(n-1)y`

where
`n` starts at 1. Because the last term in the sum is
`7y`, we deduce that there are
`n-1=7\implies n=8` terms in the entire sum, so we have

`\displaystyle\sum_(n=1)^8\bigg((14-2(n-1))x+(n-1)y\bigg)=\sum_(n=1)^8\bigg((16-2n)x+(n-1)y\bigg)`