Question:
If f is a function satisfying f(x +y) = f(x) f(y) for all x,y $element of$ N such that f(1) = 3

$sum from x equals 1 to n of space f left parenthesis x right parenthesis space equals space 120$ and , find the value of n.

Answer:

It is given that,

f (x + y) = f (x) × f (y) for all x, y ∈ N … (1)

f (1) = 3

Taking x = y = 1 in (1), we obtain

f (1 + 1) = f (2) = f (1) f (1) = 3 × 3 = 9

Similarly,

f (1 + 1 + 1) = f (3) = f (1 + 2) = f (1) f (2) = 3 × 9 = 27

f (4) = f (1 + 3) = f (1) f (3) = 3 × 27 = 81

∴ f (1), f (2), f (3), …, that is 3, 9, 27, …, forms a G.P. with both the first term and common ratio equal to 3.

It is known that, $S subscript n space equals space fraction numerator a open parentheses r to the power of n space minus space 1 close parentheses over denominator r space minus 1 end fraction$

It is given that, $sum from x equals 1 to n of space f left parenthesis x right parenthesis space equals space 120$

$therefore space 120 space equals space fraction numerator 3 open parentheses 3 to the power of n space minus 1 close parentheses over denominator 3 minus 1 end fraction rightwards double arrow 120 space equals space 3 over 2 open parentheses 3 to the power of n space minus space 1 close parentheses rightwards double arrow 3 to the power of n space minus space 1 space equals space 80 rightwards double arrow 3 to the power of n space equals space 81 space equals space 3 to the power of 4 space therefore space n space equals 4$

Thus, the value of n is 4.

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Question:
If f is a function satisfying f(x +y) = f(x) f(y) for all x,y $element of$ N such that f(1) = 3

$sum from x equals 1 to n of space f left parenthesis x right parenthesis space equals space 120$ and , find the value of n.

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Question: If f is a function satisfying f(x +y) = f(x) f(y) for all x,y N such that f(1) = 3 and , find the value of n.

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Question:
If f is a function satisfying f(x +y) = f(x) f(y) for all x,y $element of$ N such that f(1) = 3

$sum from x equals 1 to n of space f left parenthesis x right parenthesis space equals space 120$ and , find the value of n.