Question 675337: find the next term of the sequence {1,4,11,26,57,120,...}?
Found 2 solutions by Alan3354, Edwin McCravy:
Answer by Alan3354(69443)   (Show Source):
Answer by Edwin McCravy(20056)  (Show Source):
You can put this solution on YOUR website!
The other tutor just gave you the answer. He didn't show how
to get it.
1,4,11,26,57,120,...}?
a1 = 1, a2 = 4, a3 = 11, a4 = 26, ...
For a recursion formula, let's try a linear combination
of the previous term, an-1, the number of
the term, n, and a constant. We'll let the coefficients be p, q,
and the constant be r:
an = p·an-1 + q·n + r
For n = 2
an = p·an-1 + q·n + r
a2 = p·a2-1 + q·2 + r
a2 = p·a1 + q·2 + r
4 = p·1 + 2q + r
4 = p + 2q + r
For n = 3
an = p·an-1 + q·n + r
a3 = p·a3-1 + q·3 + r
a3 = p·a2 + q·3 + r
11 = p·4 + 3q + r
11 = 4p + 3q + r
For n = 4
an = p·an-1 + q·n + r
a4 = p·a4-1 + q·4 + r
a4 = p·a3 + q·4 + r
26 = p·11 + 4q + r
26 = 11p + 4q + r
So we solve this system of equations:
4 = p + 2q + r
11 = 4p + 3q + r
26 = 11p + 4q + r
and get p=2, q=1, r=0
So the recursion becomes:
an = p·an-1 + q·n + r
an = 2·an-1 + 1·n + 0
or
an = 2an-1 + n
Now we check to see if this recursion holds
for the remaining given terms:
2(11) + 4 = 22 + 4 = 26 yes
2(26) + 5 = 52 + 5 = 57 yes
2(57) + 6 = 114 + 6 = 120 yes
Indeed it does, so the next term is
2(120) + 7 = 240 + 7 = 247
Edwin
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