Question

The first three terms of an arithmetic sequence are 2k+3, 5k-2 and 10k-15, how do you show that k =4?

Answer 1

Use the definition of an arithmetic sequence to set up a system of two equations and two unknowns and solve.

Explanation:

Since the sequence is arithmetic, there is a number `d` (the "common difference") with the property that `5k-2=(2k+3)+d` and `10k-15=(5k-2)+d`. The first equation can be simplified to `3k-d=5` and the second to `5k-d=13`. You can now subtract the first of these last two equations from the second to get `2k=8`, implying that `k=4`.

Alternatively, you could have set `d=3k-5=5k-13` and solved for `k=4` that way instead of subtracting one equation from the other.

It's not necessary to find, but the common difference `d=3k-5=3*4-5=7`. The three terms in the sequence are 11, 18, 25.