The pattern 5, 11, 21, 35, … is given by the rule tn = 2n2 + 3.Show that the difference between consecutive terms, starting at the nth, is 4n + 2
Question
The pattern 5, 11, 21, 35, … is given by the rule t_n = 2n^2 + 3. Show that the difference between consecutive terms, starting at the nth, is 4n + 2.
Solution
Sure, let's break it down:
The nth term of the sequence is given by the formula tn = 2n^2 + 3.
The (n+1)th term of the sequence is given by the formula t(n+1) = 2(n+1)^2 + 3.
The difference between consecutive terms starting at the nth term is given by:
t(n+1) - tn = [2(n+1)^2 + 3] - [2n^2 + 3]
Simplify the equation:
= 2(n^2 + 2n + 1) + 3 - 2n^2 - 3 = 2n^2 + 4n + 2 - 2n^2 = 4n + 2
So, the difference between consecutive terms, starting at the nth term, is indeed 4n + 2.
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