Free Principle of Mathematical Induction 02 Practice Test - 11th Grade - Commerce
Question 1
If n ∈ N, then 11n+2 + 122n+1 is divisible by
113
123
133
None of these
SOLUTION
Solution : C
Putting n = 1 in 11n+2+122n+1
We get, 111+2+122×1+1 = 113+123 = 3059, which
is divisible by 133.
Question 2
For all positive integral values of n, 32n - 2n + 1 is
divisible by
2
4
8
12
SOLUTION
Solution : A
Putting n = 2 in 32n - 2n + 1 then,
32×2 - 2×2+1 = 81 - 4 + 1 = 78, which is divisible
by 2.
Question 3
∀ n ϵ N, P(n):2.7n+3.5n−5 is divisible by
64
676
17
24
SOLUTION
Solution : D
P(n):2.7n+3.5n−5P(1):=14+15−5=24
Suppose 24 divides P(n):2.7n+3.5n−5
P(1) is true.
Suppose P(k) is true.
2.7k+3.5k−5 is divisible by 24.
∴2.7k+3.5k−5=24m2.7k+1+3.5k+1−5=7.(2.7k)+5.(3.5k)−5=5(2.7k+3.5k−5)+2.2.7k+20=5(2.7k+3.5k−5)+4.7k+20=5(24m)+4.7k+20
Now, prove that P1(n):4.7n+20 is divisible by 24
P1(1):28+20=48→divisible by 24P1(k):4.7k+20 is divisible by 24→assume true⇒4.7k+20=24qNow, 4.7k+1+20=7.(4.7k)+20=6.4.7k+(4.7k+20)=24.7k+24qP1(k+1) is true
∴4.7n+20 is divisible by 24
∴5(24m)+4.7k+20 is divisible by 24
P(k) is true ⇒ P(k+1) is true
Hence, P(n) is true.
Question 4
∀ n ϵ N, P(n):4.7n+20 is divisible by 24. State true or false.
True
False
SOLUTION
Solution : A
P(n):4.7n+20 is divisible by 24
P(1):28+20=48→divisible by 24P(k):4.7k+20 is divisible by 24→assume true⇒4.7k+20=24qNow, 4.7k+1+20=7.(4.7k)+20=6.4.7k+(4.7k+20)=24.7k+24qP(k+1) is true
∴P(n) is true4.7n+20 is divisible by 24
Question 5
For all natural numbers n, (n+1)(n+2)(n+3) is divisible by
6
5
7
4
SOLUTION
Solution : A
P(n):(n+1)(n+2)(n+3)P(1):2×3×4=24→divisible by 6P(n):(n+1)(n+2)(n+3) is divisible by 6
P(1) is true. Assume P(k) is true
⇒(k+1)(k+2)(k+3)=6mP(k+1):(k+2)(k+3)(k+4)=(k+1)(k+2)(k+3)+3(k+2)(k+3)
Since (k+2) and (k+3) are two consecutive natural numbers, one of them must be even i.e. their product is even, say 2q
∴(k+2)(k+3)(k+4)=(k+1)(k+2)(k+3)+3.2q=6m+6q=6(m+q)→divisible by 6⇒P(k+1) is true⇒P(n) is true
Question 6
an−bn is always divisible by
2a-b
a+b
a-b
a-2b
SOLUTION
Solution : C
Substituting n=1, we get
a1−b1=a−b
Let P(n):an−bn is divisible by a-b
P(1) is true
Assume P(k) is true
ak−bk is divisible by a-bak−bk=m(a−b)Now, ak+1−bk+1=a.(ak−bk)+(a−b)bk=(am+bk)(a−b)→divisible by a-b
P(k+1) is true
Hence, P(n) is true.
Question 7
For all natural numbers n, 23n−7n−1 is divisible by
64
36
49
25
SOLUTION
Solution : C
Substitute n=1 in 23n−7n−1, we get
23−7−1=0→divisible by all positive integersSubstitute n=2 in 23n−7n−1, we get
26−14−1=49
Let P(n):23n−7n−1 is divisible by 49
P(2) is true.
Assume P(k) is true
23k−7k−1=49m
Substituting k+1 in place of n, we get
23k+3−7(k+1)−1=8.23k−7k−8=8.(23k−7k−1)+7.7k=49(8m+k)→divisible by 49
P(k+1) is true
Hence, P(n) is true.
Question 8
For all natural numbers n, 102n−1+1 is divisible by
13
11
9
none of these
SOLUTION
Solution : B
Substituting n=1, we have
101+1=11Let P(n):102n−1+1 is divisible by 11
P(1) is true.
Assume P(k) is true
⇒102k−1+1=11mNow, P(k+1):102k+1+1=100.102k−1+1=99.102k−1+102k−1+111.(9.102k−1+m)→divisible by 11
P(k+1) is also true.
Hence, P(n) is true.
Question 9
7n−3n is always divisible by
6
10
4
8
SOLUTION
Solution : C
Substituting n=1, we get
71−31=4
Let P(n):7n−3n is divisible by 4
P(1) is true
Assume P(k) is true
7k−3k is divisible by 47k−3k=4mNow, 7k+1−3k+1=7.(7k−3k)+4.3k=4.(7m+3k)→divisible by 4
P(k+1) is true
Hence, P(n) is true.
Question 10
Which of the following illustrates the inductive step to prove a statement P(n) about natural numbers n by mathematical induction, where k is an arbitrary natural number?
P(1) is true
P(k) is true
P(k) is true ⇒ P(k+1) is true
none of these
SOLUTION
Solution : C
Suppose there is a given statement P(n) involving natural numbers n such that
i) The statement is true for a specific natural number m i.e. P(m) is true. This is known as the base case.
ii) If the statement is true for n = k (where k is some positive integer), then the statement is also true for n=k+1, i.e., the truth of P(k) implies the truth of P(k+1).
The second step is called the inductive step.