show that the statements are true for all natural numbers
1^3 +2^3 +5^3+....+(2n+3)^3 =n^2(2n^2 -1)
To prove that the statement is true for all natural numbers, we can use mathematical induction.
Step 1: Base Case
Check if the statement holds true for the smallest natural number, which is 1.
1^3 = 1
n^2(2n^2 - 1) = 1^2(2(1)^2 - 1) = 1
The base case statement is true.
Step 2: Inductive Hypothesis
Assume that the statement is true for some arbitrary value of k, where k is a natural number. This is known as the inductive hypothesis.
1^3 + 2^3 + 5^3 + ... + (2k + 3)^3 = k^2(2k^2 - 1) ---- (1)
Step 3: Inductive Step
We need to prove that the statement is also true for k + 1.
1^3 + 2^3 + 5^3 + ... + (2(k + 1) + 3)^3 = (k + 1)^2(2(k + 1)^2 - 1)
We can rewrite the left side of the equation by expanding the last term:
1^3 + 2^3 + 5^3 + ... + (2k + 3)^3 + [(2(k + 1) + 3)^3] = (k + 1)^2(2(k + 1)^2 - 1)
Now, we can substitute the inductive hypothesis (equation 1) into the left side of the equation:
k^2(2k^2 - 1) + [(2(k + 1) + 3)^3] = (k + 1)^2(2(k + 1)^2 - 1)
Expanding the terms on both sides, we get:
2k^4 + 5k^3 + 4k^2 + k + 1 + 8k^3 + 36k^2 + 54k + 27 = 2k^4 + 6k^3 + k^2 + 6k + 1
Simplifying, we have:
2k^4 + 13k^3 + 40k^2 + 59k + 28 = 2k^4 + 13k^3 + 40k^2 + 59k + 28
Since both sides of the equation are equal, we have shown that if the statement is true for k, then it is also true for k + 1.
Step 4: Conclusion
By completing the base case and the inductive step, we have proven the statement to be true for all natural numbers.