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Aneli [31]
3 years ago
14

The recursive function LaTeX: f\left(0\right)=1,\:f\left(n\right)=f\left(n-1\right)+2nf ( 0 ) = 1 , f ( n ) = f ( n − 1 ) + 2 n

represents
Mathematics
1 answer:
m_a_m_a [10]3 years ago
3 0

Question:

The recursive function f(0) = 1, f(n) = f(n-1) + 2n represents the nth term of a sequence. Determine the explicit function

Answer:

f(n) = n^2 + n+1

Step-by-step explanation:

Given

f(0) = 1

f(n) = f(n-1) + 2n

Required

Write an explicit formula

Let n = 1

f(1) = f(1-1) + 2*1

f(1) = f(0) + 2

f(1) = 1 + 2 = 3

Let n = 2

f(2) = f(2-1) + 2*2

f(2) = f(1) + 4

f(2) = 3 + 4 = 7

Let n =3

f(3)=f(3-1) + 2 * 3

f(3)=f(2) + 6

f(3)=7+ 6 = 13

Let n = 4

f(4) = f(4 - 1) + 2 * 4

f(4) = f(3) + 8

f(4) = 13+ 8 = 21

So, we have:

f(1) = 3= 1 + 2 * 1=1+0*1*2*1

f(2) = 7= 3 + 2 * 2=1+1*2+2*2  

f(3) = 13= 7 + 2 * 3= 1+2*3+2*3

f(4) = 21 = 13 + 2 * 4= 1+3*4 +2 *4

Following the above pattern:

f(n) = 1 + (n - 1) * n + 2 * n

f(n) = 1 + n(n - 1) + 2n

Open bracket

f(n) = 1 + n^2 - n + 2n

f(n) = 1 + n^2 + n

f(n) = n^2 + n+1

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