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Principle of Mathematical Induction — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsPrinciple of Mathematical Induction1 Mark
MCQ
$\{1/(3 ∙ 5)\} + \{1/(5 ∙ 7)\} + \{1/(7 ∙ 9)\} + ……. + 1/\{(2n + 1) (2n + 3)\}:$
  • A
    $n/(2n + 3)$
  • B
    $n/\{2(2n + 3)\}$
  • ✓
    $n/\{3(2n + 3)\}$
  • D
    $n/\{4(2n + 3)\}$
✓

Answer

Correct option: C.
$n/\{3(2n + 3)\}$
Let the given statement be $P(n).$ Then,
$P(n): \{1/(3 ∙ 5) + 1/(5 ∙ 7) + 1/(7 ∙ 9)\} + ……. + 1/\{(2n + 1)(2n + 3)\} = n/\{3(2n + 3)\}.$
Putting $n = 1$ in the given statement, we get
and $\text{LHS} = 1/(3 ∙ 5) = 1/15$ and $\text{RHS} = 1/\{3(2 \times 1 + 3)\} = 1/15.$
$\text{LHS = RHS}$
Thus, $P(1)$ is true.
Let $P(k)$ be true. Then,
$P(k): \{1/(3 ∙ 5) + 1/(5 ∙ 7) + 1/(7 ∙ 9) + …….. + 1/\{(2k + 1)(2k + 3)\} = k/\{3(2k + 3)\} ….. (i)$
Now, $1/(3 ∙ 5) + 1/(5 ∙ 7) + ..…… + 1/[(2k + 1)(2k + 3)] + 1/[\{2(k + 1) + 1\}2(k + 1) + 3$
$= \{1/(3 ∙ 5) + 1/(5 ∙ 7) + ……. + [1/(2k + 1)(2k + 3)]\} + 1/\{(2k + 3)(2k + 5)\}$
$= k/[3(2k + 3)] + 1/[2k + 3)(2k + 5)] [$using $(i)]$
$= \{k(2k + 5) + 3\}/\{3(2k + 3)(2k + 5)\}$
$= (2k^2 + 5k + 3)/[3(2k + 3)(2k + 5)]$
$= \{(k + 1)(2k + 3)\}/\{3(2k + 3)(2k + 5)\}$
$= (k + 1)/\{3(2k + 5)\}$
$= (k + 1)/[3\{2(k + 1) + 3\}]$
$= P(k + 1): 1/(3 ∙ 5) + 1/(5 ∙ 7) + …….. + 1/[2k + 1)(2k + 3)] + 1/[\{2(k + 1) + 1\}\{2(k + 1) + 3\}]$
$= (k + 1)/\{3\{2(k + 1)\} + 3\}]$
$\Rightarrow P(k + 1)$ is true, whenever $P(k)$ is true.
Thus, $P(1)$ is true and $P(k + 1)$ is true, whenever $P(k)$ is true.
Hence, by the principle of mathematical induction, $P(n)$ is true for $n \in N$

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