A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying Sk+1 completely.
Sn: 1 + 4 + 7 + . . . + (3n - 2) = n(3n - 1)/2
I think they gave you Sk
k(3k-1)/2
Sk+1 = Sk + 3 (k+1) - 2
= (3 k^2 - k)/2 + 3 k +1
= (1/2)( 3 k^2 - k + 6 k +2 )
= (1/2)(3 k^2 + 5k + 2)
= 1.5 k^2 + 2.5 k + 1
Thank you so much !
To write the statements Sk and Sk+1 and simplify Sk+1 completely based on the given statement Sn, let's analyze the pattern in Sn:
Sn: 1 + 4 + 7 + . . . + (3n - 2) = n(3n - 1)/2
We can observe that the numbers being added in Sn have a difference of 3 between them. The first term is 1, the second term is 4 (1 + 3), the third term is 7 (4 + 3), and so on. The last term we are summing is (3n - 2).
To write Sk and simplify Sk+1, we need to find a general formula for the sum of the first k terms in the sequence.
To do this, we need to find a pattern or formula for the terms that can be used to express each term without having to list them individually.
Looking at the pattern of differences, we can see that the difference between successive terms is always 3. This indicates that the sequence is an arithmetic sequence with a common difference of 3.
The general formula for an arithmetic sequence is An = A1 + (n - 1)d, where An is the nth term, A1 is the first term, n is the index of the term, and d is the common difference.
In this case, the first term A1 = 1 and the common difference d = 3. Therefore, the nth term An can be expressed as An = 1 + (n - 1)3 = 3n - 2.
Now that we have the formula for the nth term, we can proceed to write Sk and simplify Sk+1:
Sk: 1 + 4 + 7 + . . . + (3k - 2) = k(3k - 1)/2
To simplify Sk+1, we need to calculate the sum of the first k+1 terms:
Sk+1: 1 + 4 + 7 + . . . + (3(k+1) - 2) = (k+1)(3(k+1) - 1)/2
Expanding the expression on the right side of the equation in Sk+1:
Sk+1: 1 + 4 + 7 + . . . + (3k + 3 - 2) = (k+1)(3k + 3 - 1)/2
Simplifying the expression:
Sk+1: 1 + 4 + 7 + . . . + (3k + 1) = (k+1)(3k + 2)/2
Therefore, the simplified form of Sk+1 is (k+1)(3k + 2)/2.