If f(x + y) = f(x) + f(y) + 2xy and f(1) = 3 find: f(5)…

We are given:

f(x+y)=f(x)+f(y)+2xy

and

f(1)=3

We need to find:

f(5)

—

Step 1: Find Pattern

Put:

x=y=1

Then:

f(2)=f(1)+f(1)+2(1)(1)

f(2)=3+3+2

f(2)=8

—

Step 2: Find

Using:

f(3)=f(2+1)

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

f(3)=8+3+4

f(3)=15

—

Step 3: Find

f(4)=f(3)+f(1)+2(3)(1)

f(4)=15+3+6

f(4)=24

—

Step 4: Find

f(5)=f(4)+f(1)+2(4)(1)

f(5)=24+3+8

f(5)=35

—

Final Answer

f(5)=35

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