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Principle of Mathematical Induction

Long Answer Type

21.

Using principle of mathematical induction, prove that

$cos space straight alpha space space cos space 2 straight alpha space space cos space 4 straight alpha space............... cos left parenthesis 2 to the power of straight n minus 1 end exponent straight alpha right parenthesis space equals space fraction numerator sin space 2 to the power of straight n straight alpha over denominator 2 to the power of straight n sinα end fraction$ for all $straight n space element of space straight N$

161 Views

Short Answer Type

22.
Using principle of mathematical induction, show that $straight n less than 2 to the power of straight n space space for space all space straight n space element of space straight N$

Let P(n): $space space straight n less than 2 to the power of straight n$

I. For n = 1, $straight P left parenthesis 1 right parenthesis colon space 1 space less than space 2 to the power of 1 rightwards double arrow space 1 space less than space 2$

∴ P(1) is true

II. Suppose the statement P(n) is true for n = m, $straight m space element of space straight N.$

∴     P(m) : $straight m less than 2 to the power of straight m comma space space straight m space element of space straight N$                                                  ...(i)

III.   For n = m + 1,

       $straight P left parenthesis straight m space plus space 1 right parenthesis colon space straight m space plus space 1 space less than 2 to the power of straight m plus 1 end exponent$                                              ...(ii)

       From (i), $space space straight m less than 2 to the power of straight m$

rightwards double arrow

space space 2. straight m less than 2 to the power of straight m.2 space rightwards double arrow space 2 straight m less than 2 to the power of straight m plus 1 end exponent rightwards double arrow space straight m plus straight m less than 2 to the power of straight m plus 1 end exponent

...(iii)

But

1 less or equal than straight m comma space space straight m space element of space straight N

   ... (iv)

       Adding (iii) and (iv), we get

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∴ P (m+1) is true

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

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

147 Views

23.

Using principle of mathematical induction, show that $3 to the power of straight n greater than 2 to the power of straight n$ for all $straight n space element of space straight N$

152 Views

Multiple Choice Questions

24.

If the number of terms in the expansion of $open parentheses 1 minus 2 over straight x space plus 4 over straight x squared close parentheses to the power of straight n comma straight x not equal to 0 comma$ is 28, then the sum of the coefficients of all the terms in this expansion is

64
2187
243
243

392 Views

25.

Statement − 1: For every natural number n ≥ 2 $fraction numerator 1 over denominator square root of 1 end fraction space plus space fraction numerator 1 over denominator square root of 2 end fraction space plus space..... space plus space fraction numerator 1 over denominator square root of straight n end fraction space greater than space square root of straight n$

Statement −2: For every natural number n ≥ 2, $straight n greater or equal than 2 comma space square root of straight n left parenthesis straight n plus 1 right parenthesis space end root space less than space straight n plus 1$

Statement −1 is false, Statement −2 is true
Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1
Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.
Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

125 Views

26.

If A = $open square brackets table row 1 0 row 1 1 end table close square brackets$ and I = $open square brackets table row 1 0 row 0 1 end table close square brackets$ , then which one of the following holds for all n ≥ 1, by the principle of mathematical induction

Aⁿ = n^A – (n – 1)I
Aⁿ = 2^n-1A – (n – 1)I
Aⁿ = nA + (n – 1)I
Aⁿ = nA + (n – 1)I

131 Views

27.

Let S(K) = 1 +3+5+..... (2K-1) = 3+K². Then which of the following is true?

S(1) is correct
Principle of mathematical induction can be used to prove the formula
S(K) ≠S(K+1)
S(K) ≠S(K+1)

158 Views

28.

Maximum sum of coefficient in the expansion of (1 – x sinθ + x² )ⁿ is

1
2ⁿ
3ⁿ
3ⁿ

126 Views

29.

For positive integer n, n³ + 2n is always divisible by

30.

The acceleration of a particle starting from rest moving in a straight line with uniform acceleration is 8 m/s². The time taken by the particle to move the second metre is