Question

Find the common difference and the first term in the sequence defined by `a_n=5n+2`?

Answer 1

`a_1 = 7` and common difference `d = 5`

Explanation:

`a_1 = 5(color(blue)(1))+2 = 5+2 = 7`

`a_(n+1) - a_n = (5color(blue)((n+1)) + 2) - (5color(blue)((n)) + 2)`

`color(white)(a_(n+1) - a_n) = (5n+5+2) - (5n + 2)`

`color(white)(a_(n+1) - a_n) = 5`

Actually, the standard form of the `n`th term of an arithmetic sequence is given by the formula:

`a_n = a + d(n-1)`

where `a` is the first term and `d` the common difference.

So we could have observed:

`a_n = 5n+2 = 2+5+5n-5 = color(red)(7)+color(green)(5)(n-1)`

then just identified the initial term `a = color(red)(7)`, and common difference `d = color(green)(5)` by comparison with the standard formula.