Find the explicit formula for the following sequence 14, 21, 42, 77, ......

katdoringlo

katdoringlo

Answered question

2022-09-09

Find the explicit formula for the following sequence 14, 21, 42, 77, ......

Answer & Explanation

madirans2m

madirans2m

Beginner2022-09-10Added 17 answers

Disclaimer: A finite sequence does not determine a unique formula for the terms of the sequence. The steps below assume that once we find a sequence that appears to be constant, then it will carry on being constant.
Examine sequences of differences as follows...
Write out the original sequence:
14,21,42,77
Write out the sequence of differences between pairs of adjacent terms:
7,21,35
Write out the sequence of differences of those differences:
14,14
Having reached a constant sequence, we can use the initial terms of each of these sequences as coefficients in a formula for a general term of the sequence:
a n = 14 0 ! + 7 1 ! ( n - 1 ) + 14 2 ! ( n - 1 ) ( n - 2 )
= 14 + 7 n - 7 + 7 n 2 - 21 n + 14
= 7 n 2 - 14 n + 21
= 7 ( n 2 - 2 n + 3 )
where n=1,2,3,...

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