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Principle of Mathematical Induction

Long Answer Type

11.

Prove the following by using the principle of mathematical induction for all $straight n element of space straight N.$

a + (a + d) + (a + 2d) + ...........+ [a + (n - 1)d] = $straight n over 2 left square bracket 2 straight a plus left parenthesis straight n minus 1 right parenthesis straight d right square bracket$

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12.

Prove the following by using the principle of mathematical induction for all $straight n space element of space straight N$ :

$open parentheses 1 plus 1 over 1 close parentheses open parentheses 1 plus 1 half close parentheses open parentheses 1 plus 1 third close parentheses space........ open parentheses 1 plus 1 over straight n close parentheses space equals space left parenthesis straight n space plus space 1 right parenthesis$

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13.

Prove the following by using the principle of mathematical induction for all $straight n space element of space straight N.$

$space space 1 plus 2 plus 3 plus........ plus straight n less than 1 over 8 left parenthesis 2 straight n plus 1 right parenthesis squared$

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14.
Prove the following by using the principle of mathematical induction for all $straight n space element of space straight N$ .

$space space fraction numerator 1 over denominator straight n plus 1 end fraction space plus space fraction numerator 1 over denominator straight n plus 2 end fraction space plus space.......... space plus space fraction numerator 1 over denominator 2 straight n end fraction greater than 13 over 24$

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I. For n = 2(note this step, n>1)

 $straight P left parenthesis 2 right parenthesis colon space space fraction numerator 1 over denominator 2 plus 1 end fraction plus fraction numerator 1 over denominator 2 plus 2 end fraction greater than 13 over 24$

$space space rightwards double arrow space space space 1 third space plus space 1 fourth greater than 13 over 24 space space rightwards double arrow space 7 over 12 greater than 13 over 24 space space space rightwards double arrow space 14 over 24 greater than 13 over 24$

 which is true

∴ P(n) is true for n = 2

II. Suppose the statement is true for n = m, $straight m space element of space straight N comma space space straight m greater than 1.$

$rightwards double arrow$ $space space straight P left parenthesis straight m right parenthesis space equals space fraction numerator 1 over denominator straight m plus 1 end fraction plus fraction numerator 1 over denominator straight m plus 2 end fraction plus....... plus fraction numerator 1 over denominator 2 straight m end fraction greater than 13 over 24$ .... (i)

III. For n = m + 1,

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or $fraction numerator 1 over denominator straight m plus 2 end fraction space plus space fraction numerator 1 over denominator straight m plus 3 end fraction plus....... plus fraction numerator 1 over denominator 2 straight m plus 2 end fraction greater than 13 over 24$

or $space space fraction numerator 1 over denominator straight m plus 2 end fraction plus fraction numerator 1 over denominator straight m plus 3 end fraction plus..... plus fraction numerator 1 over denominator 2 straight m end fraction plus fraction numerator 1 over denominator 2 straight m plus 1 end fraction plus fraction numerator 1 over denominator 2 straight m plus 2 end fraction greater than 13 over 24$

 Adding $fraction numerator 1 over denominator 2 straight m plus 1 end fraction plus fraction numerator 1 over denominator 2 straight m plus 2 end fraction$ on both sides of (i), we get

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$rightwards double arrow$ $fraction numerator 1 over denominator straight m plus 2 end fraction plus fraction numerator 1 over denominator straight m plus 3 end fraction plus....... plus fraction numerator 1 over denominator 2 straight m plus 2 end fraction greater than 13 over 24 plus space fraction numerator 2 straight m plus 2 plus 2 straight m plus 1 minus 4 straight m minus 2 over denominator 2 space left parenthesis 2 straight m plus 1 right parenthesis space left parenthesis straight m plus 1 right parenthesis end fraction$

$rightwards double arrow$ $fraction numerator 1 over denominator straight m plus 2 end fraction plus fraction numerator 1 over denominator straight m plus 3 end fraction plus......... plus fraction numerator 1 over denominator 2 straight m plus 2 end fraction greater than 13 over 24 plus fraction numerator 1 over denominator 2 left parenthesis straight m plus 1 right parenthesis space left parenthesis 2 straight m plus 1 right parenthesis end fraction$

 But, $fraction numerator 1 over denominator 2 space left parenthesis straight m plus 1 right parenthesis space left parenthesis 2 straight m plus 1 right parenthesis end fraction greater than 0$

∴ $space space fraction numerator 1 over denominator straight m plus 2 end fraction space plus space fraction numerator 1 over denominator straight m plus 3 end fraction plus......... plus fraction numerator 1 over denominator 2 straight m plus 2 end fraction greater than 13 over 24$

∴ P (m + 1) is true

∴ P(m) is true $rightwards double arrow$ P(m + 1) is true

Hence, P(n) is true for all $straight n element of straight N comma space straight n greater than 1.$

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15.

Prove the following by using the principle of mathematical induction for all $space space straight n element of straight N.$

n (n + 1) (n + 5) is a multiple of 3.

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16.

Prove the following by using the principle of mathematical induction for all $straight n space element of space straight N.$

$41 to the power of straight n space minus space 14 to the power of straight n$ is a multiple of 27 for all $straight n space element of space straight N.$

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17.

Prove the following by using the principle of mathematical induction for all $straight n space element of space straight N.$

$10 to the power of 2 straight n minus 1 end exponent space plus space 1$ is divisible by 11.

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18.

Prove the following by using the principle of mathematical induction for all $straight n space element of space straight N.$

$3 to the power of 2 straight n plus 2 end exponent minus 8 straight n minus 9$ is divisible by 8.