Question

Find the 13th term of the arithmetic sequence -3x – 1,42 + 4,112 + 9, ...

Answer 1

The 13th term is 81x + 59.

Explanation:

We are given the arithmetic sequence:

`\displaystle -3x -1, \, 4x +4, \, 11x + 9 \dots`

And we want to find the 13th term.

Recall that for an arithmetic sequence, each subsequent term only differ by a common difference d. In other words:

`\displaystyle \underbrace{-3x - 1}_(x_1) + d = \underbrace{4x + 4} _ {x_2}`

Find the common difference by subtracting the first term from the second:

`d = (4x+4) - (-3x - 1)`

Distribute:

`d = (4x + 4) + (3x + 1)`

Combine like terms. Hence:

`d = 7x + 5`

The common difference is (7x + 5).

To find the 13th term, we can write a direct formula. The direct formula for an arithmetic sequence has the form:

`\displaystyle x_n = a + d(n-1)`

Where a is the initial term and d is the common difference.

The initial term is (-3x - 1) and the common difference is (7x + 5). Hence:

`\displaystyle x_n = (-3x - 1) + (7x+5)(n-1)`

To find the 13th term, let n = 13. Hence:

`\displaystyle x_(13) = (-3x - 1) + (7x + 5)((13)-1)`

Simplify:

`\displaystyle \begin{aligned}x_(13) &= (-3x-1) + (7x+5)(12) \\ &= (-3x - 1) +(84x + 60) \\ &= 81x + 59 \end{aligned}`

The 13th term is 81x + 59.