Question

Example 6 Prove that
2.7" + 3.5" - 5 is divisible by 24, for all ne N.​

Answer 1

Answer:

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Answer 2

$\large\underline{\sf{Solution-}}$

Let assume that

$\rm :\longmapsto\:P(n) : {2.7}^{n} + {3.5}^{n} - 5 \: is \: divisible \: by \: 24$

where n is a natural number.

Step :- 1 For n = 1

$\rm :\longmapsto\:P(1) : {2.7}^{1} + {3.5}^{1} - 5 \:$

$\rm \: = \:14 + 15 - 5$

$\rm \: = \:29 - 5$

$\rm \: = \:24$

$\bf\implies \:P(n) \: is \: true \: for \: n = 1$

Step :- 2 Assume that P(n) is true for n = k

[ where k is some natural number ]

$\rm :\longmapsto\:P(k) : {2.7}^{k} + {3.5}^{k} - 5 \: is \: divisible \: by \: 24$

$\rm :\longmapsto\:{2.7}^{k} + {3.5}^{k} - 5 \: = 24m$

$\rm :\longmapsto\:{2.7}^{k} = 24m - {3.5}^{k} + 5 \: - - - (1)$

Step :- 3 We have to prove that P(n) is true for n = k + 1

$\rm :\longmapsto\:{2.7}^{k + 1} + {3.5}^{k + 1} - 5 \:$

$\rm \: = \:{2.7}^{k}.7 + {3.5 }^{k}.5 - 5 \:$

$\rm \: = \:(24m - {3.5}^{k} + 5).7 + 15. {5}^{k} - 5$

[ using equation (1) ]

$\rm \: = \:168m - {21.5}^{k} + 35 + 15. {5}^{k} - 5$

$\rm \: = \:168m - {6.5}^{k} + 30$

$\rm \: = \:168m - 6({5}^{k} - 5) - - - (2)$

[ Now, we have to prove that

$\red{\rm :\longmapsto\: P(k): {5}^{k} - 1 \: is \: divisible \: by \: 4}$

For k = m, Assume that P(k) is true

$\red{\rm :\longmapsto\: {5}^{m} - 1 \: is \: divisible \: by \: 4}$

$\red{\rm :\longmapsto\: {5}^{m} - 1 = 4p}$

$\red{\rm :\longmapsto\: {5}^{m} = 4p + 1}$

For k = m + 1, we have to prove that P(k) is true.

$\red{\rm :\longmapsto\: {5}^{m + 1} - 1}$

$\red{\rm :\longmapsto\: {5}^{m}.5 - 1}$

$\red{\rm :\longmapsto\: (4p + 1)5 - 1}$

$\red{\rm :\longmapsto\: 20p + 5 - 1}$

$\red{\rm :\longmapsto\: 20p + 4}$

$\red{\rm :\longmapsto\: 4(5p + 1)}$

Thus, P(k) is true by Process of Principal Induction]

So,

Equation (2) can be rewritten as

$\rm \: = \:168m - 6(4n)$

$\rm \: = \:168m - 24n$

$\rm \: = \:24(7m - n)$

$\bf\implies \:P(n) \: is \: true \: for \: n = k + 1$

Hence,

By the Process of Principal of Mathematical Induction,

$\rm :\longmapsto\:P(n) : {2.7}^{n} + {3.5}^{n} - 5 \: is \: divisible \: by \: 24$