Problem 98 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the sum. $$\sum_{j=0}^{4}(-2)^{j}$$

Step-by-Step Solution

Answer

The sum of the geometric sequence is -11.

Step 1 Identify sequence type

You must identify the type of sequence or series. Here, the sum notation and the power of j attached to -2 indicates a geometric sequence. In a geometric sequence each term after the first is found by multiplying the previous term by a fixed, non-zero number. In this case, that number is -2.

Step 2 Determine the parameters

Our series starts at j=0 and ends at j=4. The common ratio (r) is -2, which is the number being raised to the power of j. The number of terms (n) is 5 as our count starts from 0.

Step 3 Use formula to find the sum

The formula to find the sum of a geometric sequence is \(S = \frac{a(r^n - 1)}{r - 1}\) where 'a' is the first term, 'r' is the common ratio and 'n' is the total number of terms. But since our series starts from 0, it essentially means the first term of our series is 1 because -2^0 = 1. So, applying the values to the formula, we get: \(S = \frac{1((-2)^5 - 1)}{-2 - 1}\).

Step 4 Simplify the equation

Now simplify the equation: \(S = \frac{1(-32 - 1)}{-3} = \frac{33}{3}= -11\)