Question

How do you find the next number in this sequence? 3,7,14,32,58.? thank you

Answer 1

It could be `78` or anything you like...

Explanation:

Given:

`3, 7, 14, 32, 58`

First note that no sequence is determined only by its first few terms. If we are given no other information (e.g. that the sequence is arithmetic, geometric, harmonic, etc), then the following terms could be anything.

That having been said, we can attempt to match the sequence to a formula by looking at differences between consecutive terms.

Write down the initial sequence:

`color(blue)(3), 7, 14, 32, 58`

Write down the sequence of differences between consecutive terms:

`color(blue)(4), 7, 18, 26`

Write down the sequence of differences of those differences:

`color(blue)(3), 11, 8`

Write down the sequence of differences of those differences:

`color(blue)(8), -3`

Write down the sequence of differences of those differences:

`color(blue)(-11)`

Having reached a constant sequence (albeit of just one element), then we can match the given sequence with a polynomial formula using the first term of each of these sequences as coefficients:

`a_n = color(blue)(3)/(0!) + color(blue)(4)/(1!)(n-1) + color(blue)(3)/(2!)(n-1)(n-2) + color(blue)(8)/(3!)(n-1)(n-2)(n-3) + color(blue)(-11)/(4!)(n-1)(n-2)(n-3)(n-4)`

`color(white)(a_n) = 1/24 (-11n^4 + 142n^3 - 541n^2 + 890n - 408)`

Without plugging `6` into this formula to get the next term in the sequence, we can find that value by adding another `color(blue)(-11)` to the last sequence of differences and reconstructing extra terms for each of the previous sequences in turn, ending up with something like:

`3color(white)(000)7color(white)(000)14color(white)(0000)32color(white)(0000)58color(white)(000)color(red)(78)`

`color(white)(00)4color(white)(000)7color(white)(0000)18color(white)(00000)26color(white)(000)color(red)(20)`

`color(white)(0000)3color(white)(000)11color(white)(00000)8color(white)(000)color(red)(-6)`

`color(white)(000000)8color(white)(000)-3color(white)(000)color(red)(-14)`

`color(white)(0000000)-11color(white)(00)color(red)(-11)`