SOLUTION: Wants 7 th term and general expression for 4,19,58,133,256

Question 1088003: Wants 7 th term and general expression for
4,19,58,133,256
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
4,19,58,133,256
:
we solve this by taking successive levels of differences between succeeding terms - this is called "The method of common differences"
:
1) 15, 39, 75, 123
2) 24, 36, 48
3) 12, 12
:
we see that level 3 has the same difference between the 3 terms, so we have a n^3 term
:
the general form for a cubic is
:
an^3 + bn^2 + cn + d, where a,b,c,d are numbers
:
we know that
a(1^3) + b(1^2) + c(1) + d = 4
4) a + b + c + d = 4
:
a(2^3) + b(2^2) + c(2) + d = 19
5) 8a + 4b +2c + d = 19
:
a(3^3) + b(3^2) + c(3) + d = 58
6) 27a + 9b +3c +d = 58
:
a(4^3) + b(4^2) + c(4) + d = 133
7) 64a +16b +4c + d = 133
:
Equations 4 through 7 gives us 4 equations in 4 unknowns which can be solved using any n by n solver for linear systems
:
a=2, b=0, c=1, d=1
:
******************************
a(n) = 2n^3 + n + 1
:
a(7) = 2(7^3) + 7 + 1 = 694
******************************
:

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