Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# 2 + 5 + 8 + 11 + ... + (3n − 1) = 1 2 N ( 3 N + 1 ) - Mathematics

2 + 5 + 8 + 11 + ... + (3n − 1) = $\frac{1}{2}n(3n + 1)$

#### Solution

Let P(n) be the given statement.
Now,

$P(n) = 2 + 5 + 8 + . . . + (3n - 1) = \frac{1}{2}n(3n + 1)$

$\text{ Step} 1:$

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

$\text{ Hence, P(1) is true .}$

$\text{ Step} 2:$

$\text{ Let P(m) be true } .$

$\text{ Then, }$

$2 + 5 + 8 + . . . + (3m - 1) = \frac{1}{2}m(3m + 1)$

$\text{ To prove: P(m + 1) is true } .$

$\text{ That is, }$

$2 + 5 + 8 + . . . + (3m + 2) = \frac{1}{2}(m + 1)(3m + 4)$

$\text{ P(m) is equal to:}$

$2 + 5 + 8 + . . . + (3m - 1) = \frac{1}{2}m(3m + 1)$

$\text{ Thus, we have:}$

$2 + 5 + 8 + . . . + (3m - 1) + (3m + 2) = \frac{1}{2}m(3m + 1) + (3m + 2) \left[ \text{ Adding } (3m + 2) \text{ to both sides } ] \right]$

$\Rightarrow 2 + 5 + 8 + . . . + (3m + 2) = \frac{1}{2}(3 m^2 + m + 6m + 4) = \frac{1}{2}(3 m^2 + 7m + 4)$

$\Rightarrow 2 + 5 + 8 + . . . + (3m + 2) = \frac{1}{2}(3m + 4)(m + 1)$

$\text{ Thus, P(m + 1) is true } .$

$\text{ By the principle of mathematical induction, P(n) is true for all n} \in N .$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 12 Mathematical Induction
Exercise 12.2 | Q 11 | Page 27