Home
Class 12
MATHS
Let S(k)=1+3+5+.....+(2k-1)=3+k^2 . Whic...

Let `S(k)=1+3+5+.....+(2k-1)=3+k^2` . Which of the following is true ?

A

Principle of mathematical Induction can be used to prove the formula

B

S(k) implies S(k+1)

C

S(k) implies S(k+1)

D

S(1) is correct

Text Solution

Verified by Experts

The correct Answer is:
B
Let S(k) be true. Then,
1 + 3 + 5 +…+ (2k-1)=3 + `k^(2) " …(i)"`
Now,
`S(k+1)=1+3+5+...+(2k-1)+(2k+1)=(3+k^(2))+2k+1=3+(k+1)^(2)" [Using (i)]"`
implies S(k+1) is true.
Hence, S(k) implies S(k+1).
Promotional Banner

Topper's Solved these Questions

  • MATHEMATICAL INDUCTION

    OBJECTIVE RD SHARMA ENGLISH|Exercise Exercise|30 Videos

  • LIMITS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test|58 Videos

  • MATHEMATICAL REASONING

    OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test|20 Videos

Similar Questions

Explore conceptually related problems

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

If (tan3A)/(tanA)=k(k!=1) then which of the following is not true? (a) (cosA)/(cos3A)=(k-1)/2 (b) (sin3A)/(sinA)=(2k)/(k-1) (c) (cot3A)/(cotA)=1/k (d) none of these

If 12x=k is the directrix of parabola y^2 + 4y+3x=0 , then which of the following true? (A) k=25 (B) k=0 (C) k=5 (D) k=15

(2(4k+1))/(3)ge(k(6+5)-3)/(2) Which of the following correctly describes the possible values of k in the inequality above?

Let alpha,beta and gamma be the roots of the equation x^(3)-x-1=0 . If P_(k)=(alpha)^(k)+(beta)^(k)+(gamma)^(k),kge1 , then which one of the following statements is not true?

Let S_(k) , where k = 1,2 ,....,100, denotes the sum of the infinite geometric series whose first term is (k -1)/(k!) and the common ratio is (1)/(k) . Then, the value of (100^(2))/(100!) +sum_(k=2)^(100) | (k^(2) - 3k +1) S_(k)| is....

yltx+k_(1) ygt2x+k_(2) Suppose that on a coordinate plane (0, 0) is a solution to the system of inequalities given above. Which of the following conclusions about k_(1) and k_(2) must be true?

Find the value of 'k' for which the following matrices are invertible ? (i) |{:(6,k),(-2,1):}| (ii) |{:(0,k,3),(1,-2,2),(4,3,-1):}|

Let S_k,k=1, 2, …. 100 denote the sum of the infinite geometric series whose first term is (k-1)/(K!) and the common ration is 1/k then the value of (100)^2/(100!)+ Sigma_(k=1)^(100) |(k^2-3k+1)S_k| is ____________

Let A and B be two fixed points and P , another point in the plane, moves in such a way that k_(1)PA+k_(2)PB=k_(3) , where k_(1) , k_(2) and k_(3) are real constants. The locus of P is Which one of the above is not true ?

OBJECTIVE RD SHARMA ENGLISH-MATHEMATICAL INDUCTION -Exercise
  1. Let S(k)=1+3+5+.....+(2k-1)=3+k^2 . Which of the following is true ?

    Text Solution

    |

  2. AA n in N, 49^n+16n-1 is divisible by (A) 64 (B) 49 (C) 132 (D) ...

    Text Solution

    |

  3. show that n(n^2 -1), is divisible by 24 if n is an odd positive number...

    Text Solution

    |

  4. For all n in N, 7^(2n)-48n-1 is divisible by

    Text Solution

    |

  5. Prove the following by the principle of mathematical induction:\ 5^...

    Text Solution

    |

  6. For all n in N, n^(3)+2n is divisible by

    Text Solution

    |

  7. For all n in N, 4^(n)-3n-1 is divisible by

    Text Solution

    |

  8. For all n in N, 3^(3n)-26^(n)-1 is divisible by

    Text Solution

    |

  9. If n in N, then 3^(2n)+7 is divisible by

    Text Solution

    |

  10. For all n in N, 3n^(5) + 5n^(3) + 7n is divisible by

    Text Solution

    |

  11. Find the sum of first n terms of the following series: 3+7+13+21+31+ d...

    Text Solution

    |

  12. n^(th) term of the series 4+14+30+52+ .......=

    Text Solution

    |

  13. 3 + 13 + 29 + 51 + 79+… to n terms =

    Text Solution

    |

  14. Find the sum of the following series to n term: 1^3+3^3+5^3+7^3+ dot

    Text Solution

    |

  15. If 10^(n)+3*4^(n+2) + is divisible by 9, for all ninN, then the least ...

    Text Solution

    |

  16. If x^n-1 is divisible by x-k then the least positive integral value of...

    Text Solution

    |

  17. If a,b are distinct rational numbers, then for all n in N the number a...

    Text Solution

    |

  18. If n is an odd positive integer, then a^(n)+b^(n) is divisible by

    Text Solution

    |

  19. If n is an even positive integer, then a^(n)+b^(n) is divisible by

    Text Solution

    |

  20. For all n in N, (n^(5))/(5)+(n^(3))/(3)+(7n)/(15) is

    Text Solution

    |

  21. The sum of n terms of the series 1+(1+a)+(1+a+a^(2))+(1+a+a^(2)+a^(...

    Text Solution

    |