Explicit: t(n) = (n + 1)(n/2) ; Recursive:t(n) = t(n -1)+ n ; t(1) = 1,
This is a quadratic function because it takes two levels of differences.
1 3 6 10
2 3 4
1 1
To find an explicit formula [if you don’t already know this as the triangular numbers] is to look at a pattern of products.
1 = 2 x .5
3 = 3 x 1
6 = 4 x 1.5
10 = 5 x 2
the next number should be 6 x 2.5 = 15
The explicit formula is (n+1)(n/2) or (n^2 +n)/2 [The formula for triangular numbers]
For the recursive formula look at the 1st differences, 2,3,4,… the formula for this is n + 1. But you will be using the term before, or t(n-1) so (n + 1) for (n - 1) becomes just “n”
t(n)= t(n-1) + n
You know the 4th term to find the 5th term, take the 4th term and add 5
10 + 5 = 15
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