Leaving Cert Mathematics Induction (Proof): Prove using induction that, for

Prove using induction that, for all n ∈ ℕ, the sum of the first n square numbers can be found using the formula: $$1^2 + 2^2 + 3^2 + .. - Leaving Cert Mathematics - Question d - 2020

Question d

Prove-using-induction-that,-for-all-n-∈-ℕ,-the-sum-of-the-first-n-square-numbers-can-be-found-using-the-formula:--$$1^2-+-2^2-+-3^2-+-..-Leaving Cert Mathematics-Question d-2020.png

Prove using induction that, for all n ∈ ℕ, the sum of the first n square numbers can be found using the formula: $$1^2 + 2^2 + 3^2 + ... + n^2 = \frac{n(n+1)(2n+1)}... show full transcript

Worked Solution & Example Answer:Prove using induction that, for all n ∈ ℕ, the sum of the first n square numbers can be found using the formula: $$1^2 + 2^2 + 3^2 + .. - Leaving Cert Mathematics - Question d - 2020

Step 1

Step P(1):

96%

114 rated

Answer

To prove the base case, for n = 1:

$LHS = 1^2 = 1$

Using the formula:

$RHS = \frac{1(1+1)(2 \cdot 1 + 1)}{6} = \frac{1 \cdot 2 \cdot 3}{6} = 1$

Thus, for n = 1, LHS = RHS.

Step 2

Step P(k):

99%

104 rated

Answer

Assume the formula holds for n = k:

$1^2 + 2^2 + 3^2 + ... + k^2 = \frac{k(k+1)(2k+1)}{6}$

Step 3

Step P(k + 1):

96%

101 rated

Answer

Now prove it for n = k + 1:

$LHS = 1^2 + 2^2 + 3^2 + ... + k^2 + (k+1)^2$

Using the induction hypothesis:

$= \frac{k(k+1)(2k+1)}{6} + (k+1)^2$

Combine the terms:

$= \frac{k(k+1)(2k+1) + 6(k+1)^2}{6}$

Factor out (k + 1):

$= \frac{(k+1)(k(2k+1) + 6(k+1))}{6}$

Simplifying further:

$= \frac{(k+1)(2k^2 + 7k + 6)}{6}$

Recognizing that (2k^2 + 7k + 6 = (k+2)(2k+3)$$, we can write:

$LHS = \frac{(k+1)(k+2)(2k+3)}{6} = RHS$

Step 4

Concluding statement:

98%

120 rated

Answer

Therefore, by mathematical induction, the formula is true for all natural numbers n.

Prove using induction that, for all n ∈ ℕ, the sum of the first n square numbers can be found using the formula: $$1^2 + 2^2 + 3^2 + .. - Leaving Cert Mathematics - Question d - 2020

Worked Solution & Example Answer:Prove using induction that, for all n ∈ ℕ, the sum of the first n square numbers can be found using the formula: $$1^2 + 2^2 + 3^2 + .. - Leaving Cert Mathematics - Question d - 2020

Step P(1):

Step P(k):

Step P(k + 1):

Concluding statement:

Join the Leaving Cert students using SimpleStudy...