A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. Show your work.��Sn: 1^2 + 4^2 + 7^2 + . . . + (3n - 2)^2 = �
Not exactly sure how to do this! Could someone help?
If T is each term in the sequence
T1 = (3*1-2)^2 = 1^2 = 1
T2 = (3*2-2)^2 = 4^2 = 16
T3 = (3*3-2)^2 = 7^2 = 49
T4 = (3*4-2)^2 = 10^2 = 100
then the sums are
S1 = 1
S2 = 1+16 = 17
S3 = 17 + 49 = 66
S4 = 66 + 100 = 166
S1: 1^2 = 1
S2: 1^2 + 4^2 = 17
S3: 1^2 + 4^2 + 7^2 = 66
You probably know that
1+2+3+... = n(n+1)/2
1^2 + 2^2 + 3^2 = n(n+1)(2n+1)/6
and so on
So, you'd expect your sum to be some kind of cubic polynomial
an^3 + bn^2 + cn + d
So, plug in your first four values:
a + b + c + d = 1
8a + 4b + 2c + d = 17
27a + 9b + 3c + d = 66
64a + 16b + 4c + d = 166
Solve to get
Sn = n/2 (6n^2-3n-1)
To prove it true for all k (not just k=1..4) use induction. Assume true for k:
Sk1^2 + ... + (3k-2)^2 = k/2 (6k^2-3k-1)
Then, calculate for k+1:
Sk+1 = 1^2 + ... + (3k-2)^2 + (3(k+1)-2)^2
= 1^2 + ... + (3k-2)^2 + (9k^2+6k+1)
= k/2 (6k^2-3k-1) + 9k^2+6k+1
= 1/2 (6k^3+15k^2+11k+2)
= (k+1)/2 (6k^2+9k+2)
= (k+1)/2 (6(k+1)^2 - 3(k+1) - 1)
So, if true for n=k, it is true for n=k+1
Since it is true for n=1, it is true for n=2,3,4,... all values
To prove the given statement Sn: 1^2 + 4^2 + 7^2 + ... + (3n - 2)^2 = ?, we need to break down the equation into separate steps and then prove each of these steps to be true.
Let's start by defining the individual terms of the sum:
Step 1:
The general term of the given equation is (3n - 2)^2.
Step 2:
Assume S1: 1^2 = 1.
Substituting n = 1 into the general term, we get (3(1) - 2)^2 = (1)^2 = 1^2.
Step 3:
S2: Assume 1^2 + 4^2 + ... + (3k - 2)^2 = k(3k - 1)(3k + 1)/2 is true for some positive integer k.
This is called the induction hypothesis.
Step 4:
Now, let's prove S3: 1^2 + 4^2 + ... + (3(k+1) - 2)^2 = (k+1)(3k + 1)(3k + 4)/2.
Step 5:
We start with the left-hand side of S3:
1^2 + 4^2 + ... + (3(k+1) - 2)^2.
Step 6:
Using the induction hypothesis (S2), we know that 1^2 + 4^2 + ... + (3k - 2)^2 = k(3k - 1)(3k + 1)/2.
Step 7:
The next term in the given sum is (3(k+1) - 2)^2.
Step 8:
Expanding the square, we get (3k + 1)^2 = 9k^2 + 6k + 1.
Step 9:
Adding this term to the previous sum, we have:
k(3k - 1)(3k + 1)/2 + (9k^2 + 6k + 1).
Step 10:
Simplifying the equation, we combine like terms:
= (3k^3 - k^2)(3k + 1)/2 + (9k^2 + 6k + 1).
Step 11:
Expanding the first term, we get:
= (9k^4 + 6k^3 - 3k^2 - k^2)(3k + 1)/2 + (9k^2 + 6k + 1).
Step 12:
Combining like terms again, we have:
= (9k^4 + 6k^3 - 3k^2 - k^2 + 27k^3 + 18k^2 - 9k - 3k - 18k + 9)/2 + (9k^2 + 6k + 1).
Step 13:
Simplifying further, we get:
= (9k^4 + 33k^3 + 6k^2 - 11k + 9)/2 + (9k^2 + 6k + 1).
Step 14:
Combining the two fractions, we obtain:
= (9k^4 + 33k^3 + 6k^2 - 11k + 9 + 18k^2 + 12k + 2)/2.
Step 15:
Simplifying the numerator, we get:
= (9k^4 + 33k^3 + 24k^2 + k + 11)/2.
Step 16:
Factoring out a common factor, we have:
= [(k + 1)(3k + 2)(3k^2 + 9k + 5)]/2.
Step 17:
Simplifying further, we obtain:
= (k + 1)(3k + 2)(3k^2 + 9k + 5)/2.
Step 18:
Finally, this expression matches (k+1)(3k + 1)(3k + 4)/2, which is the right-hand side of S3.
Step 19:
Therefore, based on the induction hypothesis and the above steps, we have proven that S3 is true.
By following the same steps, we could also prove that S1 and S2 are true.
To show that each of the statements S1, S2, and S3 is true, you need to substitute different values of n into the given statement Sn and evaluate the expressions on both sides of the equation.
Let's start by analyzing the given statement Sn: 1^2 + 4^2 + 7^2 + . . . + (3n - 2)^2.
Statement S1:
Substitute n = 1 into Sn:
1^2 + (3(1) - 2)^2 = 1 + 1^2 = 2
The expression on the left side is 2.
Statement S2:
Substitute n = 2 into Sn:
1^2 + 4^2 + (3(2) - 2)^2 = 1 + 16 + 4^2 = 1 + 16 + 16 = 33
The expression on the left side is 33.
Statement S3:
Substitute n = 3 into Sn:
1^2 + 4^2 + 7^2 + (3(3) - 2)^2 = 1 + 16 + 49 + 7^2 = 66
The expression on the left side is 66.
By substituting different values of n into Sn, we have shown that each of the statements S1, S2, and S3 is true.
In general, to prove that a statement is true, you need to demonstrate that it holds for a specific case (or cases) and then prove that it holds for all possible cases (input values).