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

Gauge Odom

Gauge Odom

Answered question

2022-09-06

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

Answer & Explanation

Nodussimj

Nodussimj

Beginner2022-09-07Added 14 answers

Each of the last 3 terms looks close to a cube, so I could guess the formula as a n = n 3 - 2 , but let's pretend I didn't spot that...
Write out the initial sequence:
−1,6,25,62,123
Write out the sequence of differences of that sequence:
7,19,37,61
Write out the sequence of differences of that sequence:
12,18,24
Write out the sequence of differences of that sequence:
6,6
Having reached a constant sequence, we can now use the first number of each sequence to write out a formula for a n as follows:
a n = - 1 0 ! + 7 1 ! ( n - 1 ) + 12 2 ! ( n - 1 ) ( n - 2 ) + 6 3 ! ( n - 1 ) ( n - 2 ) ( n - 3 )
= - 1 + 7 ( n - 1 ) + 6 ( n - 1 ) ( n - 2 ) + ( n - 1 ) ( n - 2 ) ( n - 3 )
= - 1 + 7 n - 7 + 6 n 2 - 18 n + 12 + n 3 - 6 n 2 + 11 n - 6
= n 3 - 2
So a 20 = 20 3 - 2 = 8000 - 2 = 7998

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