Question

17.
Use Mathematical induction to prove:

4 + 9 +14 +…..+(5n-1) = ( n/2) (5n+3)

Answer 1

Answer: plse use the given formula in rs aggarwal book

Step-by-step explanation:

Answer 2

To prove by mathematical induction, let Pₙ be the equation:

Let Pₙ be 4 +9 +14 +...+(5n -1)= (n/2)(5n +3).

1) Base step: prove P₁ is true

When n=1,

LHS= 4

RHS

$= \frac{1}{2} (5 + 3)$

$= \frac{1}{2} (8)$

= 4

= LHS

∴ P₁ is true.

2) Assume Pₙ is true,

i.e. 4 +9 +14 +...+(5n -1)= (n/2)(5n +3)

3) Induction step: prove Pₙ₊₁ is true

Find Pₙ₊₁ by substituting (n+1) into n on both sides.

Pₙ₊₁: 4 +9 +14 +...+(5n -1)+ [5(n+1) -1]=

$( \frac{n + 1}{2} )[5(n + 1) + 3]$

Prove that the LHS is equal to the RHS.

LHS

= 4 +9 +14 +...+(5n -1)+ [5n +5-1]

$= \bf{4 +9 +14 +...+(5n -1)}+ (5n +4)$

Substitute the value of Pₙ in (2) into bolded expression:

$= \frac{n}{2} (5n + 3) + 5n + 4$

$= \frac{5}{2} n^{2} + \frac{3}{2}n + 5n + 4$

$= \frac{1}{2} (5 {n}^{2} + 3n + 10n + 8)$

$= \frac{1}{2} (5 {n}^{2} + 13n+ 8)$

$= \frac{1}{2} (n + 1)(5n + 8)$

$= \frac{n + 1}{2} (5n + 8)$

$= \frac{n + 1}{2} [5(n + 1) + 3]$

= RHS

(shown)

4) Write concluding statements

∴ If Pₙ is true, Pₙ₊₁ is true

Since P₁ is true, Pₙ is true for all positive integers of n.

To learn more about mathematical induction, do check out the following!

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