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PRINCIPLE OF MATHEMATICAL INDUCTION — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSPRINCIPLE OF MATHEMATICAL INDUCTION1 Mark
MCQ
If p(n): $49^\text{n}+16^{\text{n}}\lambda$ is divisible by 64 for $\text{n}\in\text{N}$ is true, then the least negative integral value of $\lambda$ is:
  • A
    -3
  • B
    -2
  • ✓
    -1
  • D
    -4
✓

Answer

Correct option: C.
-1
  1. -1
Solution:
$(49)^n+16 n-1$
$\Rightarrow(1+48)^n+16 n-1$
$\Rightarrow 1+48 n+\ldots 48^n+16 n-1$
$\Rightarrow 64 n+n C_2(48)^2+n C_3(48)^3+\ldots+(48)^n$
$\Rightarrow 64\left(n+n C_2(6)^2+n C_3(6)^3 48+\ldots+(6)^n 8^{n-2}\right)$
$\therefore 49^n+16 n-1 \text { is divisible by } 64$

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