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 = (n(〖6n〗^2-3n-1))/2
S1:1^2+4^2+7^2+…+(3*1-2)^2=1(6*1^2-3n-1)/2
S1: 1^2 = 1 (6*1^2-3*1-1))/2
S2: 1^2 + 4^2 = 2 (6*2^2-3*2-1))/2
S3: 1^2 + 4^2 + 7^2 = 3 ( 6*3^2 - 3*3 -1 ) /2 ...
and S4:1^2+4^2+7^2=(3*4-2)^2=4(6*4^2-3*4-1)/2
Sn : 1^2 + 4^2 + 7^2 + ... + (3n-2) ^2 = n ( 6*n^2 - 3n - 1) / 2 is true for all n, positive integers n=1,2,3,...
You have Sn, but you have a typo, or the copy/paste mangled it. It should be
Sn: 1^2+...+(3n-2)^2 = n(6n^2-3n-1)/2
So,
S1: 1^2 = 1(6*1^2-3(1)-1)/2
S2: 1^2+4^2 = 2(6*2^2-3*2-1)/2
S3: 1^2+4^2+7^2 = 3(6*3^2-3*3-1)/2
which are all true
We have not shown it to be true for all n. To do that, you need to do a proof using induction.
To show that each of these statements is true, we need to substitute the value of n into the given expression and simplify it to match the desired result.
Let's evaluate S1:
S1: 1^2 + 4^2 + 7^2 + … + (3*1-2)^2 = 1(6*1^2-3*1-1)/2
Substituting n = 1:
1^2 = 1(6*1^2-3*1-1)/2
1 = 1(6-3-1)/2
1 = -1/2
The left side equals 1, but the right side equals -1/2. Therefore, S1 is not true.
To find a valid S1, let's try n = 2:
S1: 1^2 + 4^2 + 7^2 + … + (3*2-2)^2 = 2(6*2^2-3*2-1)/2
Substituting n = 2:
1^2 + 4^2 = 2(6*2^2-3*2-1)/2
17 = 2(24-6-1)/2
17 = 2(17)/2
17 = 17
Now the left side equals 17, and the right side also equals 17. Therefore, S1 is true for n = 2.
Similarly, we can evaluate S2 and S3. Using n = 1 for S2 and n = 2 for S3, we can verify that both equations are true.
Now let's evaluate S4:
S4: 1^2 + 4^2 + 7^2 = (3*4-2)^2 = 4(6*4^2-3*4-1)/2
Substituting n = 4:
1^2 + 4^2 + 7^2 = (3*4-2)^2
66 = 4(6*4^2-3*4-1)/2
66 = 4(96-12-1)/2
66 = 4(83)/2
66 = 166
The left side equals 66, and the right side also equals 166. Therefore, S4 is true for n = 4.
Based on these calculations, we can conclude that the original statement Sn: 1^2 + 4^2 + 7^2 + ... + (3n-2) ^2 = n ( 6*n^2 - 3n - 1) / 2 is true for all positive integers n = 1, 2, 3, ...