What is the sum of a 17–term arithmetic sequence where the first term is 7 and the last term is –105?
To find the sum of a finite arithmetic sequence, you can use the formula:
Sn = (n/2)(a1 + an)
where Sn represents the sum of the sequence, n is the number of terms, a1 is the first term, and an is the last term.
In this case, we are given:
- First term (a1) = 7
- Last term (an) = -105
We need to find the number of terms (n).
To find the number of terms (n), we can use the formula for the nth term of an arithmetic sequence:
an = a1 + (n-1)d
where d represents the common difference between the terms of the arithmetic sequence.
Rearranging the formula, we have:
n = (an - a1 + d) / d
In this case, we know:
- first term (a1) = 7
- last term (an) = -105
Now the only thing left to find is the common difference (d), which we can calculate using the formula:
d = (an - a1) / (n - 1)
Substituting the values we know, we can calculate:
d = (-105 - 7) / (n - 1)
Simplifying the equation, we have:
d = -112 / (n - 1)
Now we can substitute this value of d into the equation we derived earlier to find n:
n = (an - a1 + d) / d
n = (-105 - 7 + (-112 / (n - 1))) / (-112 / (n - 1))
To solve this equation, we can cross-multiply and simplify. After finding the value of n, we can use the formula Sn = (n/2)(a1 + an) to find the sum of the arithmetic sequence.