Question

How do I find this, what’s the formula to find the 20th term and can anyone explain the formula? Will award brainliest

Answer 1

Answer:

• f(n) = n^2 +1
• f(20) = 401

Step-by-step explanation:

The first differences of the sequence are ...

• 5-2 = 3
• 10-5 = 5
• 17-10 = 7
• 26-17 = 9
• 37-26 = 11

Second differences are ...

• 5 -3 = 2
• 7 -5 = 2
• 9 -7 = 2
• 11 -9 = 2

The second differences are constant, so the sequence can be described by a second-degree polynomial.

We can write and solve three equations for the coefficients of the polynomial. Let's define the polynomial for the sequence as ...

  f(n) = an^2 + bn + c

Then the first three terms of the sequence are ...

• f(1) = 2 = a·1^2 + b·1 + c
• f(2) = 5 = a·2^2 +b·2 + c
• f(3) = 10 = a·3^2 +b·3 +c

Subtracting the first equation from the other two gives ...

  3a +b = 3

  8a +2b = 8

Subtracting the first of these from half the second gives ...

  (4a +b) -(3a +b) = (4) -(3)

  a = 1 . . . . . simplify

Substituting into the first of the 2-term equations, we get ...

  3·1 +b = 3

  b = 0

And substituting the values for a and b into the equation for f(1), we have ...

  1·1 + 0 + c = 2

  c = 1

So, the formula for the sequence is ...

  f(n) = n^2 + 1

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The 20th term is f(20):

  f(20) = 20^2 +1 = 401

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Comment on the solution

It looks like this matches the solution of the "worked example" on your problem page.