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

2 Answers
Oct 11, 2017

5050

Explanation:

Let a_1=2a1=2, a_2=5a2=5, a_3=10a3=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

Oct 11, 2017

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.