Home
Class 12
MATHS
A sequence a(1),a(2),a(3), . . . is defi...

A sequence `a_(1),a_(2),a_(3), . . .` is defined by letting `a_(1)=3` and `a_(k)=7a_(k-1)`, for all natural numbers `kge2`. Show that `a_(n)=3*7^(n-1)` for natural numbers.

Text Solution

Verified by Experts

We have a sequence `a_(1),a_(2),a_(3)"…."` is defined by letting `a_(1) = 3` and `a_(k) = 7a_(k-1)`, for all natural number `k ge 2`.
Let `P(n) : a_(n) = 3 xx 7^(2-1) = 3 xx 7^(1) = 21`
Also, `a_(1) = 3, a_(k) = 7a_(k-1)`
`rArr a_(2) = 7a_(1) =a = 7 xx 3 = 21`
Thus, `P(2)` is true.
Now, assume that `P(k)` is true.
`:. a_(k)= 3 xx 7^(k-1)`
Now, to prove `P(k+1)` we have to show that
`a_(k+1) = 3 xx 7^(k+1-1)`
Given that `a_(k) = 7a_(k-1)`
So, `a_(k+1) = 7a_(k+1-1)`
`= 7a_(k)`
`= 7 xx 3 xx 7^(k-1)`
`= 3 xx 7^((k+1)-1)`
Hence, `P(k+1)` is true whenever `P(k)` is true.
So, by the principle of mathematical inducton, `P(n)` is true for any natural number `n`.
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • PERMUTATION AND COMBINATION

    CENGAGE PUBLICATION|Exercise Comprehension|8 Videos

  • PROBABILITY

    CENGAGE PUBLICATION|Exercise All Questions|470 Videos

Similar Questions

Explore conceptually related problems

a_(1), a_(2),a_(3) in R - {0} and a_(1)+ a_(2)cos2x+ a_(3)sin^(2)x=0 " for all " x in R then

A sequence of no. a_1,a_2,a_3 ..... satisfies the relation a_n=a_(n-1)+a_(n-2) for nge2 . Find a_4 if a_1=a_2=1 .

Knowledge Check

  • If positive integers a_(1), a_(2), a_(3),… are ion A.P. such that a_(8) +a_(10) =24, then the value of a_(9) is-

    A
    10
    B
    11
    C
    12
    D
    9

    • Similar Questions

    Explore conceptually related problems

    Let a_(0)=0 and a_(n)=3a_(n-1)+1 for n ge 1 . Then the remainder obtained dividing a_(2010) by 11 is

    If a_(1), a_(2), a_(3), …., a_(n) are in H.P., prove that, a_(1)a_(2) + a_(2)a_(3) + a_(3)a_(4) +…+ a_(n-1)a_(n) = (n-1)a_(1)a_(n)

    If A_(1), A_(2),..,A_(n) are any n events, then

    Suppose a_(1), a_(2) , .... Are real numbers, with a_(1) ne 0 . If a_(1), a_(2), a_(3) , ... Are in A.P., then

    Show that a_(1), a_(2) ,……, a_(n) …. Form an AP where a_(n) is defined as below: (i) a_(n) = 3 + 4n , (ii) a_(n) = 9-5n Also find the sum of the first 15 terms in each case.

    If a_(1) = 2 and a_(n) - a_(n-1) = 2n (n ge 2) , find the value of a_(1) + a_(2) + a_(3)+…+a_(20) .

    The Fibonacci sequence is defined by 1=a_(1)=a_(2) and a_n =a_(n-1)+a_(n-2) , n gt 2 Find (a_(n+1))/a_n for n = 1, 2, 3, 4, 5

    Home
    Profile