# How do you find the explicit formula and calculate term 20 for -1, 6, 25, 62, 123?

Look at sequences of differences to construct a formula and find the

Put the first sequence in writing:

List the differences in that sequence in writing:

List the differences in that sequence in writing:

List the differences in that sequence in writing:

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The given sequence is: -1, 6, 25, 62, 123.

To find the explicit formula, we can look for patterns in the differences between consecutive terms.

1st difference: 6 - (-1) = 7 2nd difference: 25 - 6 = 19 3rd difference: 62 - 25 = 37 4th difference: 123 - 62 = 61

We notice that the differences are not constant, but they seem to follow a pattern. The differences themselves form a sequence of odd numbers (7, 19, 37, 61), which are consecutive odd integers starting from 7.

This suggests that the original sequence might be generated by a cubic function, as the differences between consecutive terms are quadratic.

Let's assume the explicit formula is of the form $f(n) = an^3 + bn^2 + cn + d$.

Given that the first term is -1, we can find the values of a, b, c, and d by plugging in the values of n and the corresponding terms:

When $n = 1$: $a(1)^3 + b(1)^2 + c(1) + d = -1$ $a + b + c + d = -1$ ...(1)

When $n = 2$: $a(2)^3 + b(2)^2 + c(2) + d = 6$ $8a + 4b + 2c + d = 6$ ...(2)

When $n = 3$: $a(3)^3 + b(3)^2 + c(3) + d = 25$ $27a + 9b + 3c + d = 25$ ...(3)

When $n = 4$: $a(4)^3 + b(4)^2 + c(4) + d = 62$ $64a + 16b + 4c + d = 62$ ...(4)

Solving the system of equations (1)-(4) will give us the values of a, b, c, and d.

Once we have the explicit formula, we can calculate the 20th term by substituting $n = 20$ into the formula and evaluating it.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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