The sum of the first "n" terms of series is 2^(n+3) -8.Find the 7th term.
Assuming a geometric series,
Sn = a(r^n-1)/(r-1)
So, since
2^(n+3)-8 = 2^n*2^3 - 8 = 8*2^n-8 = 8(2^n-1)
Looks like a=8 and r=2.
S7 = 8*2^6 = 512
To find the 7th term of the given series, we need to determine the value of "n" that represents the 7th term.
Given that the sum of the first "n" terms of the series is given by the formula:
Sum(n) = 2^(n+3) - 8
We can use this formula to find the value of "n" when the sum is equal to the sum of the first 6 terms.
So, let's find the sum of the first 6 terms:
Sum(6) = 2^(6+3) - 8
= 2^9 - 8
= 512 - 8
= 504
Since we know that the series can be expressed as the sum of the first 6 terms, we can set it equal to 504 and solve for "n":
504 = 2^(n+3) - 8
Adding 8 to both sides of the equation:
512 = 2^(n+3)
To solve for "n", take the logarithm of both sides with base 2:
log2(512) = log2(2^(n+3))
Simplifying the right side using the logarithmic property:
log2(512) = (n+3) * log2(2)
Since log2(2) equals 1, we can simplify further:
log2(512) = n + 3
Now, solve for "n":
n = log2(512) - 3
Calculating:
n = 9 - 3
n = 6
So, the value of "n" representing the 7th term is 6.
Now, we can find the 7th term of the series by substituting "n" into the series formula:
Term(7) = 2^(n+3) - 8 = 2^(6+3) - 8
= 2^9 - 8
= 512 - 8
= 504
Therefore, the 7th term of the given series is 504.