11: Prove the following by using the principle of mathematical induction for all n€ N:
1/1.2.3 + 1/2.3.4 + 1/3.4.5 + … +1/n(n+1)(n+2) = n(n+3)/4(n+1)(n+2)
Solution:
Let the given statement be P(n), i.e.,
P(n): 1/1.2.3 + 1/2.3.4 + 1/3.4.5 + … +1/n(n+1)(n+2) = n(n+3)/4(n+1)(n+2)
For n = 1, we have
P(1): 1/1.2.3 = 1.(1+3)/4(1+1)(1+2) = 1.4/4.2.3 = 1/1.2.3, which is true.
Let P(k) be true for some positive integer k, i.e.,
1/1.2.3 + 1/2.3.4 + 1/3.4.5 + … +1/k(k+1)(k+2) = k(k+3)/4(k+1)(k+2)
We shall now prove that P(k + 1) is true.
Consider
Thus, P(k + 1) is true whenever P(k) is true.
Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.
- Prove the following by using the principle of mathematical induction for all n€ N:
a + ar + ar2+ …+arn-1 = {a(rn– 1)} / {(r – 1)}
Solution:
Let the given statement be P(n), i.e.,
P(n): a + ar + ar2+ …+arn-1 = {a(rn– 1)} / {(r – 1)}
For n = 1, we have ,
P(1): a = {a(r1– 1)} / {(r – 1)} = a which is true.
Let P(k) be true for some positive integer k, i.e.,
a + ar + ar2+ …+arn-1 = {a(rn– 1)} / {(r – 1)} ————(1)
We shall now prove that P(k + 1) is true.
Consider
Thus, P(k + 1) is true whenever P(k) is true.
Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.
13: Prove the following by using the principle of mathematical induction for all n € N:
(1+ 3/1) (1+ 5/4) (1+ 7/9)… (1+ (2n+1)/n2) = (n+1)2
Solution:
Let the given statement be P(n), i.e.,
P(n): (1+ 3/1) (1+ 5/4) (1+ 7/9)… (1+ (2n+1)/n2) = (n+1)2
For n = 1, we have
P(1): (1 + 3/1) = 4 = (1+1)2 = 22 = 4, which is true
Let P(k) be true for some positive integer k, i.e.,
(1+ 3/1) (1+ 5/4) (1+ 7/9)… (1+ (2k+1)/k2) = (k+1)2
We shall now prove that P(k + 1) is true.
Consider
Thus, P(k + 1) is true whenever P(k) is true.
Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.
- Prove the following by using the principle of mathematical induction for all n € N:
(1 + 1/1)(1 + 1/2)(1 + 1/3)…(1 + 1/n) = (n + 1)
Solution:
Let the given statement be P(n), i.e.,
P(n): (1 + 1/1)(1 + 1/2)(1 + 1/3)…(1 + 1/n) = (n + 1)
For n = 1, we have ,
P(1): (1 + 1/1) = 2 = (1+1) which is true.
Let P(k) be true for some positive integer k, i.e.,
P(k): (1 + 1/1)(1 + 1/2)(1 + 1/3)…(1 + 1/k) = (k + 1) ————-(1)
We shall now prove that P(k + 1) is true.
Consider
Thus, P(k + 1) is true whenever P(k) is true. Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.
15: Prove the following by using the principle of mathematical induction for all n € N:
12 + 32 + 52 + … +(2n – 1)2 = n(2n-1)(2n+1)/3
Solution:
Let the given statement be P(n), i.e.,
P(n): 12 + 32 + 52 + … +(2n – 1)2 = n(2n-1)(2n+1)/3
For n = 1, we have
P(1) = 12 = 1 = 1(2.1-1)(2.1+1)/3 = 1.1.3/3 = 1, which is true.
Let P(k) be true for some positive ineteger k, i.e.,
P(k) = 12 + 32 + 52 + … +(2k – 1)2 = k(2k-1)(2k+1)/3 ———(1)
We shall now prove that P(k + 1) is true.
Consider
Thus, P(k + 1) is true whenever P(k) is true. Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.
16: Prove the following by using the principle of mathematical induction for all n€ N:
1/1.4 + 1/4.7 + 1/7.10 + … + 1/(3n-2)(3n+1) = n/(3n+1)
Solution:
Let the given statement be P(n), i.e.,
P(n): 1/1.4 + 1/4.7 + 1/7.10 + … + 1/(3n-2)(3n+1) = n/(3n+1)
For n = 1, we have
P(1): 1/1.4 = 1/3.1+1 = 1/4 = 1/1.4, which is true.
Let P(k) be true for some positive integer k, i.e.,
P(k): 1/1.4 + 1/4.7 + 1/7.10 + … + 1/(3k-2)(3k+1) = k/(3k+1) ———-(1)
We shall now prove that P(k+1) is true.
Consider
Thus, P(k + 1) is true whenever P(k) is true. Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.
- Prove the following by using the principle of mathematical induction for all n € N:
1/3.5 + 1/5.7 + 1/7.9 + … + 1/(2n+1)(2n+3) = n/3(2n+3)
Solution:
Let the given statement be P(n), i.e.,
1/3.5 + 1/5.7 + 1/7.9 + … + 1/(2n+1)(2n+3) = n/3(2n+3)
For n = 1, we have ,
P(1): 1/3.5 = 1/3(2.1+3) = 1/3.5 which is true.
Let P(k) be true for some positive integer k, i.e.,
1/3.5 + 1/5.7 + 1/7.9 + … + 1/(2k+1)(2k+3) = k/3(2k+3) ————(1)
We shall now prove that P(k + 1) is true.
Consider
Thus, P(k + 1) is true whenever P(k) is true.
Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.
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