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: 2 + 5 + 8 + . . . + ( 3n - 1) = n(1 + 3n)/2
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S(1) = 2
by the formula: S(1) = 1(1 + 3)/2 = 2
S(2) = 2+5 = 7
by the formula: S(2) = 2(1+6)/2 = 7
S(3) = 2+5+8 = 15
by formula: S(3) = 3(1+9)/2 = 15
seems to work.
To show that it is true for any n, use your knowledge of some sums (k=1 to n):
sum(1) = n
sum(k) = n(n+1)/2
Now just simplify
3*sum(k) - sum(1)
To prove that each of the statements S1, S2, and S3 is true, we need to substitute different values of n into the statement Sn and verify that the equation holds true.
S1: For n = 1
If we substitute n = 1 into Sn, we get:
2 + 5 + 8 + . . . + (3(1) - 1) = 1(1 + 3(1))/2
2 + 5 + 8 + . . . + 2 = 1(1 + 3)/2
2 + 2 = 2
4 = 4
Since the left side of the equation is equal to the right side, S1 is true.
S2: For n = 2
If we substitute n = 2 into Sn, we get:
2 + 5 + 8 + . . . + (3(2) - 1) = 2(1 + 3(2))/2
2 + 5 + 8 + . . . + 5 = 2(1 + 6)/2
2 + 5 = 7
7 = 7
Since the left side of the equation is equal to the right side, S2 is true.
S3: For n = 3
If we substitute n = 3 into Sn, we get:
2 + 5 + 8 + . . . + (3(3) - 1) = 3(1 + 3(3))/2
2 + 5 + 8 + . . . + 8 = 3(1 + 9)/2
2 + 5 + 8 = 15
15 = 15
Since the left side of the equation is equal to the right side, S3 is true.
By showing that the statement holds true for specific values of n, we have proven that S1, S2, and S3 are true.