If P n is the statement -2- - -3n- and If P r is true- prove that P r -1 is true-

P(n) : 2^n ≤ 3n Given that P(r) is true ⇒ 2^r ≤ 3r Multiplying both the sides by 2. 2.2^r ≤ 2.3r 2^2+1≤ 6r ≤ 3r + 3r ≤ 3 + 3r. [Since 3r ≤ 3 ⇒ 3r + 3r ≤ 3 + 3r] ...

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Standard XII

Mathematics

Proof by mathematical induction

If P n is the...

Question

If P (n) is the statement "2" $\leq$ "3n", and If P (r) is true, prove that P (r + 1) is true.

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Solution

$p (n) : 2^{n} \leq 3 n$

Given that P(r) is true

$\Rightarrow 2^{r} \leq 3 r$

Multiplying both the sides by 2.

${2.2}^{r} \leq 2.3 r$

$2^{2 + 1} \leq 6 r$

$\leq 3 r + 3 r$

$\leq 3 + 3 r$ .

$[S i n c e 3 r \leq 3 \Rightarrow 3 r + 3 r \leq 3 + 3 r]$

$2^{r + 1} \leq 3 (r + 1)$

$\Rightarrow$ P(r + 1) is true.

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If P (n) is the statement "2" ≤ "3n", and If P (r) is true, prove that P (r + 1) is true.

p(n):2n≤3n Given that P(r) is true ⇒2r≤3r Multiplying both the sides by 2. 2.2r≤2.3r 22+1≤6r ≤3r+3r ≤3+3r. [Since 3r≤3⇒3r+3r≤3+3r] 2r+1≤3(r+1) ⇒ P(r + 1) is true.

If P (n) is the statement "2" $\leq$ "3n", and If P (r) is true, prove that P (r + 1) is true.