Verified Solution • 2025–26 Session moderate

QJEEM23Maed7e: JEE Main 2023 Mathematics - Class 12 JEEM23Maed7e Solution: Suppose be a differentiable function such that . If , then ∑ n…

Suppose f:R→(0,∞)f: \mathbb{R} \rightarrow(0, \infty) be a differentiable function such that 5f(x+y)=f(x)⋅f(y),∀x,y∈R5 f(x+y)=f(x) \cdot f(y), \forall x, y \in \mathbb{R} . If f(3)=320f(3)=320 , then ∑n=05f(n) is equal to :

Multiple Choice Options

  • A 6875
  • B 6525
  • C 6575
  • D 6825

Step-by-Step Verified Solution

5f(x+y)=f(x).f(y)5f(x + y) = f(x).f(y)

5f(3)=f(1).f(2)5f(3) = f(1).f(2)

5f(2)=(f(1))25f(2) = {(f(1))^2}

f(10)=5f(10) = 5

f(1)=20f(1) = 20

⇒f(1).(f(1))25=1600\Rightarrow f(1).{{{{(f(1))}^2}} \over 5} = 1600

∑n=05f(n)=f(0)+20+80+320+1280+5120\sum\limits_{n = 0}^5 {f(n) = f(0) + 20 + 80 + 320 + 1280 + 5120}

=1750+5120=6825= 1750 + 5120 = 6825

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