1-2-5-1-5-8-1-8-11- -1-3 n-13 n-2-n-6 n-4

Let P(n) : 1/2.5 + 1/5.8 + 1/8.11 + ...... + 1/(3n 1)(3n + 2) = n/6bn + 4 Put n = 1 P(1) : 1/2.5 = 1/6 + 4 1/10 = 1/10 ⇒P(n) is true for n = 1 Let P(n) is tr ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Proof by mathematical induction

1/2.5+1/5.8+1...

Question

$\frac{1}{2.5} + \frac{1}{5.8} + \frac{1}{8.11} + . . . . + \frac{1}{(3 n - 1) (3 n + 2)} = \frac{n}{6 n + 4}$

Open in App

Solution

Let P(n) : $\frac{1}{2.5} + \frac{1}{5.8} + \frac{1}{8.11} + . . . . . . + \frac{1}{(3 n - 1) (3 n + 2)} = \frac{n}{6 b n + 4}$

Put n = 1

P(1) : $\frac{1}{2.5} = \frac{1}{6 + 4}$

$\frac{1}{10} = \frac{1}{10}$

$\Rightarrow$ P(n) is true for n = 1

Let P(n) is true for n = k, so

$\frac{1}{2.5} + \frac{1}{5.8} + \frac{1}{8.11} + . . . . . . . + \frac{1}{(3 k - 1) (3 k + 2)} = \frac{k}{(6 k + 4)}$ ..... (1)

We have to show that,

$\frac{1}{2.5} + \frac{1}{5.8} + \frac{1}{8.11} + . . . . . + \frac{1}{(3 k - 1) (3 k + 2)} + \frac{1}{(3 k + 2) (3 k + 5)} = \frac{(k + 1)}{(6 k + 10)}$

Now,

${\frac{1}{2.5} + \frac{1}{5.8} + \frac{1}{8.11} + \frac{1}{(3 k - 1) (3 k + 2)}} + \frac{1}{(3 k + 2) (3 k + 5)}$

$= \frac{k}{6 k + 4} + \frac{1}{(3 k + 2) (3 k + 5)}$

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

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

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

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

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

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

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

$\Rightarrow$ P(n) is true for n = k + 1

$\Rightarrow$ P(n) is true for all n $ϵ$ N by PMI

Suggest Corrections

Similar questions

$\frac{1}{2.5} + \frac{1}{5.8} + \frac{1}{8.11} + \dots + \frac{1}{(3 n - 1) (3 n + 2)} = \frac{n}{6 n + 4}$

Q. Prove the following:

\frac{1}{2.5} + \frac{1}{5.8} + \frac{1}{8.11} + \dots + \frac{1}{(3 n - 1) (3 n + 2)} = \frac{n}{6 n + 4}

Prove the following by using the principle of mathematical induction for all n ∈ N:

Q. Prove the following by using the principle of mathematical induction for all

n \in N : \frac{1}{2.5} + \frac{1}{5.8} + \frac{1}{8.11} + . . . . . . + \frac{1}{(3 n - 1) (3 n + 2)} = \frac{n}{(6 n + 4)}

Q. If

\frac{1}{2.5} + \frac{1}{5.8} + \frac{1}{8.11} + . . . . . . . . . . . . . . + \frac{1}{(3 n - 1) (3 n + 2)} = \frac{n}{a n + 4} .

Find

a

Join BYJU'S Learning Program

12.5+15.8+18.11+....+1(3n−1)(3n+2)=n6n+4

$\frac{1}{2.5} + \frac{1}{5.8} + \frac{1}{8.11} + . . . . + \frac{1}{(3 n - 1) (3 n + 2)} = \frac{n}{6 n + 4}$