Asequence is defined by the recursive function f(n  +  1)  =  –10f(n). if f(1) = 1, what is f(3)?

100



Explanation:

First term
f(1) = 1
which is given to us

We will use the equation f(n+1) = -10*f(n) to generate successive terms after this first term.

Second term
f(n+1) = -10*f(n)
f(1+1) = -10*f(1) ... plug in n = 1
f(2) = -10*f(1)
f(2) = -10*1 ... replace f(1) with 1 (works because f(1) = 1)
f(2) = -10

Put another way, the second term is -10 times the first term
second term = -10*(first term)
second term = -10*(1)
second term = -10

Third term
f(n+1) = -10*f(n)
f(2+1) = -10*f(2) ... replace n with 2
f(3) = -10*f(2)
f(3) = -10*(-10) ... replace f(2) with -10 (works because f(2) = -10)
f(3) = 100 which is the answer we want

In other words, the third term is -10 times the second term
third term = -10*(second term)
third term = -10*(-10)
third term = 100

Summary of the three terms
f(1) = 1
f(2) = -10
f(3) = 100
Each term is found by multiplying the previous term by -10

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