If Sn = 1 + 6 + 11 + 16 + 21 + 26...... where an = (5n − 4), what would be the mathematical expression represented by Sk + ak + 1?
A) (k(5k − 3)/2)+ 5k + 1
B) (k(5k − 3)/2)+ 5k − 4
C) (k(5k − 3)/2)+ 5k + 2
D) (k(5k − 3)/2)+ 5k − 3
E) (k(5k − 3)/2)+ 5k − 8
as with any arithmetic progression,
Sk = k/2(a1 + ak)
= k/2(1+5k-4)
= k/2(5k-3)
Sk+ak+1 = k(5k-3)/2 + 5k-4+1
(D)
Don't forget your algebra I just because you're in pre-calc now.
To find the mathematical expression represented by Sk + ak + 1, we need to substitute the values of Sk and ak + 1 into the equation and simplify.
Given that Sn = 1 + 6 + 11 + 16 + 21 + 26... and an = (5n − 4), we can see that the common difference between each term is 5. Therefore, the nth term of the arithmetic sequence Sn can be expressed as Sn = 1 + (n-1)*5.
Now, let's find the value of Sk. The kth term (Sk) can be found by substituting k into the formula Sn = 1 + (n-1)*5. So, Sk = 1 + (k-1)*5 = 5k - 4.
Next, let's find the value of ak + 1. The (k + 1)th term (ak + 1) can be found by substituting (k + 1) into the formula an = (5n − 4). So, ak + 1 = (5(k+1) - 4) = 5k + 1.
Now, let's substitute the values of Sk and ak + 1 into the expression Sk + ak + 1 and simplify:
Sk + ak + 1 = (5k - 4) + (5k + 1) = 10k - 3
So, the mathematical expression represented by Sk + ak + 1 is 10k - 3.
Therefore, the correct answer is D) (k(5k − 3)/2) + 5k − 3.