Question

10. Find the n" term of odd natural number
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Answer 1

General form of an even natural number is, n = 2q + 1, q ≥ 0

Let this equation generate a sequence in ascending order, with each n as the terms

Then, the reqd. sequence is: 1, 3, 5, 7, 9, 11, 13, ...

Let $a_n$ represent nth term in the sequence

then first term $a_1$ = 1

Let $n_1$ and $n_2$ be any two consecutive even natural number,

$n_1 = 2k, \: n_2 = 2(k+1) + 1$

difference between $n_2$ and $n_1$ is $n_2 - n_1 = 2k+2 +1-2k-1=2$

(⇒ n2 comes after n1 in the above sequence)

Then, difference between any two consecutive even number, $\Delta = 2$

$\implies a_2 = a_1 + \Delta\\\implies a_3 = a_2 + \Delta\\\implies a_4 = a_3 + \Delta$

and so on

$\implies a_n = a_{n-1} + \Delta,\:\:\:for \:n-1 \geq 1\\ = a_{n-2} + 2\Delta\\= a_{n-3}+ 3\Delta\\ = a_{n-4}+ 4\Delta$

And so on,

$\implies a_n = a_{n-(n-1)} + (n-1)\Delta\\= a_1 + (n-1)\Delta\\= 1 + (n-1)2\\\therefore a_n=2n-1$