What is the expression, in terms of n, for the sequence:4, 13, 26, 43

The given sequence is: 4, 13, 26, 43

To find the expression for the sequence, we need to find the difference between consecutive terms:

13 - 4 = 9 26 - 13 = 13 43 - 26 = 17

The differences between the terms are not constant, so this is not an arithmetic sequence. However, the differences between the differences are:

13 - 9 = 4 17 - 13 = 4

Since the second differences are constant, this is a quadratic sequence. The second difference in a quadratic sequence is equal to 2a, where a is the coefficient of n^2 in the nth term formula. So, in this case, a = 4/2 = 2.

The general form of the nth term of a quadratic sequence is an^2 + bn + c. We already know that a = 2, so we need to find b and c.

Substituting n = 1 into the formula gives the first term of the sequence, which is 4. So:

21^2 + b1 + c = 4 2 + b + c = 4 b + c = 2

Substituting n = 2 into the formula gives the second term of the sequence, which is 13. So:

22^2 + b2 + c = 13 8 + 2b + c = 13 2b + c = 5

Subtracting the first equation from the second gives:

2b + c - (b + c) = 5 - 2 b = 3

Substituting b = 3 into the equation b + c = 2 gives:

3 + c = 2 c = -1

So the nth term of the sequence is 2n^2 + 3n - 1.