JEE Main 2026 Jan 22 Shift 2, Mathematics Q19: Arithmetic, Geometric And Harmonic Progressions

JEE Main2026Jan 22, Shift 2Mathematics

Let $f$ and $g$ be functions satisfying $f (x + y) = f (x) f (y)$ , $f (1) = 7$ and $g (x + y) = g (x y)$ , $g (1) = 1$ , for all $x, y \in N$ . $x = 1 \sum n (\frac{f ( x )}{g ( x )}) = 19607$ , then $n$ is equal to :

Solution

$f (x + y) = f (x) f (y)$ with $f (1) = 7$ gives $f (x) = 7^{x}$ .

$g (x + y) = g (x y)$ with $y = 1$ gives $g (x + 1) = g (x)$ , so $g$ is constant. Since $g (1) = 1$ , $g (x) = 1$ for all $x$ .

Therefore $x = 1 \sum n 7^{x} = 7 \cdot \frac{7 ^{n} - 1}{6} = 19607$ .

$7^{n} - 1 = \frac{6 \cdot 19607}{7} = 16806 \Rightarrow 7^{n} = 16807 = 7^{5}$ , so $n = 5$ .

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JEE Main 2026 Jan 22 Shift 2, Mathematics Q19: Arithmetic, Geometric And Harmonic Progressions

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