Question

Find the 7th term of the sequence: 2, 5, 10, . . .?

Answer 1

`50`

Explanation:

Let `a_1=2`, `a_2=5`, `a_3=10`

These numbers are increasing by increasingly odd numbers which is a hint that this sequence's rule is quadratic (a variable gets squared).

I also noticed that each of these numbers is one more than a square number `(1,4,9...)`

So the rule is `a_n=n^2+1`

Thus `a_7=7^2+1=49+1=50`

The seventh term in the sequence is `50`

Answer 2

It could be `50`, `55`, `58`, `215` or anything really.

Explanation:

Any finite number of terms does not determine the following terms, unless you are told something about the type of sequence, e.g. arithmetic, geometric, etc.

In this example, the numbers `2, 5, 10` do not form an arithmetic or geometric sequence, but we can try to find other patterns...

It could be a quadratic sequence with general formula:

`a_n = n^2+1`

in which case the first few terms are:

`2, 5, 10, 17, 26, 37, 50,...`

It could be the sum of an arithmetic and geometric sequence with general formula:

`a_n = 2^n+n-1`

in which case the first few terms are:

`2, 5, 10, 19, 29, 41, 55,...`

It could be the sequence of the sums of the first `n` primes, in which case it starts:

`2, 5, 10, 17, 28, 41, 58,...`

It could be powers of `root(3)(10)` rounded to the nearest integer:

`2, 5, 10, 22, 46, 100, 215,...`

Actually the following terms could be anything you like.