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Let a(0)=2,a1=5 and for n ge 2, an=5a(n-...

Let `a_(0)=2,a_1=5` and for `n ge 2, a_n=5a_(n-1)-6a_(n-2)`. Then prove by induction that `a_(n)=2^(n)+3^(n) forall n in Z^+`.

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To prove that \( a_n = 2^n + 3^n \) for all \( n \in \mathbb{Z}^+ \) using mathematical induction, we will follow the three steps of induction: base case, induction hypothesis, and induction step. ### Step 1: Base Case We need to verify the statement for the smallest positive integer, \( n = 0 \) and \( n = 1 \). - For \( n = 0 \): \[ a_0 = 2 ...
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ARIHANT MATHS ENGLISH-MATHEMATICAL INDUCTION -Exercise (Subjective Type Questions)
  1. Prove the following by the principle of mathematical induction:\ 11...

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  2. n^7-n is divisible by 42 .

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  3. Prove that 3^(2n)+24n-1 is divisible by 32 .

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  4. prove using mathematical induction:-n(n+1)(n+5) is divisible by 6 for ...

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  5. Prove that (25)^(n+1)-24n+5735 is divisible by (24)^2 for all n=1,2,.....

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  6. Prove the following by the principle of mathematical induction: \ x...

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  7. Prove by induction that if n is a positive integer not divisible by 3,...

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  8. Prove that the product of three consecutive positive integers is divis...

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  9. Prove by induction that the sum of the cubes of three consecutive n...

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  10. When the square of any odd number, greater than 1, is divided by 8, ...

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  11. Prove the following by using induction for all n in N. 1+2+3+.....+n=...

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  12. Prove the following by the principle of mathematical induction: 1^2...

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  13. Prove the following by the principle of mathematical induction: \ 1...

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  14. Prove the following by the principle of mathematical induction:1/(2...

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  15. Prove 1.4.7+2.5.8+3.6.9+....... upto n terms =(n)/(4)(n+1)(n+6)(n+7)

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  16. 1^2/(1.3)+2^2/(3.5)+3^2/(5.7)+.....+n^2/((2n-1)(2n+1))=((n)(n+1))/((2(...

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  17. Let a(0)=2,a1=5 and for n ge 2, an=5a(n-1)-6a(n-2). Then prove by indu...

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  18. If a(1)=1,a(n+1)=(1)/(n+1)a(n),a ge1, then prove by induction that a(n...

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  19. if a,b,c,d,e and f are six real numbers such that a+b+c=d+e+f a^2+b^2...

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  20. Prove that tan^(- 1)(1/3)+tan^(- 1)(1/7)+tan^(- 1)(1/13)+..........+ta...

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