Advertisements
Advertisements
Question
12 + 22 + 32 + ... + n2 =\[\frac{n(n + 1)(2n + 1)}{6}\] .
Advertisements
Solution
Let P(n) be the given statement.
Now,
\[P(n) = 1^2 + 2^2 + 3^2 + . . . + n^2 = \frac{n(n + 1)(2n + 1)}{6}\]
\[\text{ Step } 1: \]
\[P(1) = 1^2 = \frac{1(1 + 1)(2 + 1)}{6} = \frac{6}{6} = 1\]
\[\text{ Hence, P(1) is true} . \]
\[\text{ Step } 2: \]
\[\text{ Let P(m) be true .} \]
\[\text{ Then,} \]
\[ 1^2 + 2^2 + . . . + m^2 = \frac{m(m + 1)(2m + 1)}{6}\]
\[\text{ We shall now prove that P(m + 1) is true} . \]
\[i . e . , \]
\[ 1^2 + 2^2 + 3^2 + . . . + (m + 1 )^2 = \frac{(m + 1)(m + 2)(2m + 3)}{6}\]
\[ \text{ Now } , \]
\[P(m) = 1^2 + 2^2 + 3^2 + . . . + m^2 = \frac{m(m + 1)(2m + 1)}{6}\]
\[ \Rightarrow 1^2 + 2^2 + 3^2 + . . . + m^2 + (m + 1 )^2 = \frac{m(m + 1)(2m + 1)}{6} + (m + 1 )^2 \left[ \text{ Adding} (m + 1 )^2 \text{ to both sides} \right]\]
\[ \Rightarrow 1^2 + 2^2 + 3^2 + . . . + (m + 1 )^2 = \frac{m(m + 1)(2m + 1) + 6(m + 1 )^2}{6} = \frac{(m + 1)(2 m^2 + m + 6m + 6)}{6} = \frac{(m + 1)(m + 2)(2m + 3)}{6}\]
\[\text{ Hence, P(m + 1) is true } . \]
\[\text{ By the principle of mathematical induction, the given statement is true for all n } \in N .\]
APPEARS IN
RELATED QUESTIONS
Prove the following by using the principle of mathematical induction for all n ∈ N:
`1 + 3 + 3^2 + ... + 3^(n – 1) =((3^n -1))/2`
Prove the following by using the principle of mathematical induction for all n ∈ N:
`1^3 + 2^3 + 3^3 + ... + n^3 = ((n(n+1))/2)^2`
Prove the following by using the principle of mathematical induction for all n ∈ N:
1.2 + 2.3 + 3.4+ ... + n(n+1) = `[(n(n+1)(n+2))/3]`
Prove the following by using the principle of mathematical induction for all n ∈ 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)`
Prove the following by using the principle of mathematical induction for all n ∈ N:
Prove the following by using the principle of mathematical induction for all n ∈ N: 32n + 2 – 8n– 9 is divisible by 8.
Prove the following by using the principle of mathematical induction for all n ∈ N: 41n – 14n is a multiple of 27.
\[\frac{1}{1 . 2} + \frac{1}{2 . 3} + \frac{1}{3 . 4} + . . . + \frac{1}{n(n + 1)} = \frac{n}{n + 1}\]
\[\frac{1}{1 . 4} + \frac{1}{4 . 7} + \frac{1}{7 . 10} + . . . + \frac{1}{(3n - 2)(3n + 1)} = \frac{n}{3n + 1}\]
1.3 + 2.4 + 3.5 + ... + n. (n + 2) = \[\frac{1}{6}n(n + 1)(2n + 7)\]
a + (a + d) + (a + 2d) + ... (a + (n − 1) d) = \[\frac{n}{2}\left[ 2a + (n - 1)d \right]\]
(ab)n = anbn for all n ∈ N.
n(n + 1) (n + 5) is a multiple of 3 for all n ∈ N.
Prove that n3 - 7n + 3 is divisible by 3 for all n \[\in\] N .
Prove that 1 + 2 + 22 + ... + 2n = 2n+1 - 1 for all n \[\in\] N .
\[\text{ Let } P\left( n \right) \text{ be the statement } : 2^n \geq 3n . \text{ If } P\left( r \right) \text{ is true, then show that } P\left( r + 1 \right) \text{ is true . Do you conclude that } P\left( n \right)\text{ is true for all n } \in N?\]
Prove that the number of subsets of a set containing n distinct elements is 2n, for all n \[\in\] N .
\[\text { A sequence } x_1 , x_2 , x_3 , . . . \text{ is defined by letting } x_1 = 2 \text{ and } x_k = \frac{x_{k - 1}}{k} \text{ for all natural numbers } k, k \geq 2 . \text{ Show that } x_n = \frac{2}{n!} \text{ for all } n \in N .\]
Prove by method of induction, for all n ∈ N:
3 + 7 + 11 + ..... + to n terms = n(2n+1)
Prove by method of induction, for all n ∈ N:
12 + 22 + 32 + .... + n2 = `("n"("n" + 1)(2"n" + 1))/6`
Prove by method of induction, for all n ∈ N:
(cos θ + i sin θ)n = cos (nθ) + i sin (nθ)
Answer the following:
Given that tn+1 = 5tn − 8, t1 = 3, prove by method of induction that tn = 5n−1 + 2
Answer the following:
Prove by method of induction 152n–1 + 1 is divisible by 16, for all n ∈ N.
Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:
`sum_(t = 1)^(n - 1) t(t + 1) = (n(n - 1)(n + 1))/3`, for all natural numbers n ≥ 2.
Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:
2n + 1 < 2n, for all natual numbers n ≥ 3.
A student was asked to prove a statement P(n) by induction. He proved that P(k + 1) is true whenever P(k) is true for all k > 5 ∈ N and also that P(5) is true. On the basis of this he could conclude that P(n) is true ______.
Prove the statement by using the Principle of Mathematical Induction:
For any natural number n, 7n – 2n is divisible by 5.
A sequence d1, d2, d3 ... is defined by letting d1 = 2 and dk = `(d_(k - 1))/"k"` for all natural numbers, k ≥ 2. Show that dn = `2/(n!)` for all n ∈ N.
Prove that, cosθ cos2θ cos22θ ... cos2n–1θ = `(sin 2^n theta)/(2^n sin theta)`, for all n ∈ N.
If 10n + 3.4n+2 + k is divisible by 9 for all n ∈ N, then the least positive integral value of k is ______.
For all n ∈ N, 3.52n+1 + 23n+1 is divisible by ______.
If P(n): 2n < n!, n ∈ N, then P(n) is true for all n ≥ ______.
