recurrence - finding the rth term of a sequence -


the question give possible formula rth term. i'm able solve 2 questions rest can't seems of different way or weird.as i'm studying alevels think there's common rule or maybe easy way solve sequence related problems.i never understood sequence enough-it's hard me.

6  18  54   162

i'm able solve 2*3^r

4  7    12   19

by r^2+3

but 

4   12  24  40  60

i'm trying many ways can't find answer.i think there's common rule solving these not marks there should solved in easy way i'm not getting how to.please help

here's formula in r sequence:

g <- function(n) 6*n + 2*n^2 + 4 g(0:4) [1]  4 12 24 40 60

here 1 way solve relation. first, recognize quadratic difference arithmetic sequence (linear).

then note g(x + 1) = g(x) + 8 + 4x. represent g(x) = a*x^2 + b*x + c.

then:

g(x+1) = a(x+1)^2 + b(x+1) + c = g(x) + 8 + 4x = a*x^2 + b*x + c + 8 * 4x      ax^2 + 2ax + + b*x + b + c = a*x^2 + b*x + c + 8 + 4x

thus

           2ax +       +b      =                   8 + 4x

as holds x, must 2ax = 4x or a = 2. thus

            4x + 2 + b = 8 + 4x

so b = 6. these known, c determined g(0) = c = 4.


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