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Principle of Mathematical Induction — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsPrinciple of Mathematical Induction1 Mark
MCQ
For all $n \in N, 3n^5+ 5n^3+ 7n$ is divisible by:
  • A
    $5$
  • ✓
    $15$
  • C
    $10$
  • D
    $3$
✓

Answer

Correct option: B.
$15$
Given number $= 3n^5+ 5n^3+ 7n$
Let $n = 1, 2, 3, 4, ……..$
$3n^5+ 5n^3+ 7n$
$= 3 \times 1^2+ 5 \times 1^3+ 7 \times 1$
$= 3 + 5 + 7 $
$= 15$
$3n^5+ 5n^3+ 7n$
$= 3 \times 2^5+ 5 \times 2^3+ 7 \times 2$
$= 3 \times 32 + 5 \times 8 + 7 \times 2$
$= 96 + 40 + 14$
$= 15 \times 10$
$= 150$
$3n^5+ 5n^3+ 7n$
$= 3 \times 3^5+ 5 \times 3^3+ 7 \times 3$
$= 3 \times 243 + 5 \times 27 + 7 \times 3$
$= 729 + 135 + 21$
$= 15 \times 59$
​​​​​​​$= 885$
Since, all these numbers are divisible by $15$ for $n = 1, 2, 3, …..$
So, the given number is divisible by $1$

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