How many terms are required in the series (-6) + (-12) + (-24)... to add to a sum of -378?
How do I do this?
looks like a geometric series where
a = -6 and r = 2
Term(n) = a(r^n - 1)/(r-1), where n is the number of terms
-378 = -6(2^n - 1)/(2-1)
63 = 2^n - 1
64 = 2^n
I know that 2^6 = 64
so n=6
so you need 6 terms.
Thank you so much, could you please help me with my other calculus posts.
To find the number of terms required in the given series to add up to a sum of -378, you can use the formula for the sum of a geometric series:
Sum = first term * (1 - common ratio^n) / (1 - common ratio)
In this case, the first term is -6 and the common ratio is -2. We need to solve for the exponent 'n' in order to find the number of terms.
Here's how you can calculate it step by step:
1. Substitute the given values into the formula:
-378 = -6 * (1 - (-2)^n) / (1 - (-2))
2. Simplify the equation:
-378 = -6 * (1 + 2^n) / 3
3. Multiply both sides of the equation by 3 to eliminate the fraction:
-1134 = -6 * (1 + 2^n)
4. Divide both sides of the equation by -6:
189 = 1 + 2^n
5. Subtract 1 from both sides of the equation:
188 = 2^n
6. Take the logarithm of both sides of the equation, using any base you prefer:
log(188) = log(2^n)
7. Apply the logarithm property to bring down the exponent:
n * log(2) = log(188)
8. Divide both sides of the equation by log(2):
n = log(188) / log(2)
9. Use a calculator to evaluate the right side to find the exponent:
n ≈ 7.5754
Since the number of terms cannot be fractional, round up to the nearest whole number to get the minimum number of terms required. In this case, you would need at least 8 terms for the series to add up to a sum of -378.