help me pleaseee

Question

Consider the sequence below:

-2, 1, 6, 13, 22, …

What explicit expression can be used to find the nth term of this sequence

Answer 1

Hi Christina :-) Between two consecutive terms, you have a number that's increasing by 2 each time, starting with 3. So between two terms, you really have each time the next consecutive odd number (3, 5, 7, 9, etc.). The recurrence relation would be:

u(0) = -2

u(n) = u(n - 1) + (2 n + 1)

To get to an explicit expression, we therefore need to add odd numbers together. The sum of the k first odd numbers is k² but in this case we skip 1 and start with 3. So the direct expression would be:

u(n) = (n + 1)² - 3

I hope that makes sense.

Answer 2

The difference between each number is increasing by 2: 3, 5, 7, 9.

starting with n = 0 for -2,

difference for n=1 is 3, n = 2 is 5, n = 3 is 7, so difference is 2*n + 1

so a(n) = a(n-1) + 2*n + 1

a(1) = a(0) + 2(1) + 1 = -2 + 3 = 1

a(2) = a(1) + 2(2) + 1 = 1 + 5 = 6

a(3) = a(2) + 2(3) + 1 = 6 + 7 = 13

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