Find the sum of a finite geometric sequence from n = 1 to n = 7, using the expression −4(6)n − 1.
To find the sum of a finite geometric sequence, we can use the formula:
S = a(1 - r^n) / (1 - r)
where:
S = sum of the sequence
a = first term of the sequence
r = common ratio of the sequence
n = number of terms in the sequence
In your case, the expression for the nth term of the sequence is −4(6)n − 1. First, let's identify the values for a, r, and n.
a = −4(6)^1 − 1 = −4(6) - 1 = -24 - 1 = -25
r = −4(6) = -24
n = 7
Now we can substitute these values into the formula and calculate the sum:
S = (-25)(1 - (-24)^7) / (1 - (-24))
To simplify the calculation, let's first evaluate the exponent:
(-24)^7 = -45,435,424
Now we can substitute this value back into the formula:
S = (-25)(1 - (-45,435,424)) / (1 - (-24))
Let's simplify the numerator first:
1 - (-45,435,424) = 1 + 45,435,424 = 45,435,425
Now we can substitute this value back into the formula:
S = (-25)(45,435,425) / (1 - (-24))
Next, let's simplify the denominator:
1 - (-24) = 1 + 24 = 25
Now we can substitute this value back into the formula:
S = (-25)(45,435,425) / 25
Finally, we can perform the calculation:
S = -45,435,425
Therefore, the sum of the finite geometric sequence from n = 1 to n = 7 using the expression −4(6)n − 1 is -45,435,425.
I suspect a notational problem.
Why the -4(6)n?
Is there supposed to be an exponent, as in 2^3=8?
Do you mean -4*6^(n-1) ?
That is, the terms are
-4,24,-144,...
Anyway, just plug your numbers into the formula for the sum of a geometric sequence.
Sn = a (1-r^n)/(1-r)