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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Which answer describes the calculations that could be used to solve this problem?
Bonita brought 3 bags of apples and 2 pounds of grapes to the picnic. Each bag of apples weighed 4 pounds. How many pounds of apples and grapes did Bonita bring to the picnic?
you know that summing from 1 to n:
sum(1) = n
sum(n) = n(n+1)/2
so, sum(3n-1) = 3sum(n)-sim(1)
= 3n(n+1)/2 - n
= (3n^2+3n-2n)/2
= (3n^2+n)/2
= n(3n+1)/2
To show that each of the statements S1, S2, and S3 is true, we need to substitute different values of n into Sn and verify that the equation holds true. Let's begin:
S1: Substitute n = 1 into Sn
2 + 5 + 8 + ... + (3(1) - 1) = 1(1 + 3(1))/2
2 + 5 + 8 = 1(1 + 3)/2
15 = 4/2
15 = 2
Since 15 is not equal to 2, this equation is not true. Therefore, S1 is not true.
S2: Substitute n = 2 into Sn
2 + 5 + 8 + ... + (3(2) - 1) = 2(1 + 3(2))/2
2 + 5 + 8 + 11 = 2(1 + 6)/2
26 = 16/2
26 = 8
Again, 26 is not equal to 8, so this equation is not true. Therefore, S2 is not true.
S3: Substitute n = 3 into Sn
2 + 5 + 8 + ... + (3(3) - 1) = 3(1 + 3(3))/2
2 + 5 + 8 + 11 + 14 = 3(1 + 9)/2
40 = 30/2
40 = 15
Once again, 40 is not equal to 15, so this equation is not true. Therefore, S3 is not true.
Since none of S1, S2, or S3 are true, it is not possible to show that the statement Sn is true based on these examples. It's important to note that the original statement Sn may still be true, but S1, S2, and S3 are not valid examples for proving it.