SELECT * FROM question_mgmt as q WHERE id=1063 AND status=1 SELECT id,question_no,question,chapter FROM question_mgmt as q WHERE courseId=2 AND subId=6 AND chapterId=80 and ex_no='5' AND status=1 ORDER BY CAST(question_no AS UNSIGNED)

Question:

If f is a function satisfying f(x +y) = f(x) f(y) for all x,y element ofN  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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