Evaluate the Summation sum from k=1 to 4 of (-3/7)^k

Problem

(∑_k=1^4)(−3/7)

Solution

1. Identify the terms of the summation by substituting k=1,2,3,4 into the expression (−3/7)k

2. Calculate each individual term by raising the fraction to the corresponding power.

(a_1)=(−3/7)1=−3/7

(a_2)=(−3/7)2=9/49

(a_3)=(−3/7)3=−27/343

(a_4)=(−3/7)4=81/2401

3. Find a common denominator to add the fractions. The least common multiple of 7, 49, 343, 2401i*s()401$.

4. Convert each term to have the denominator 2401

−3/7=−1029/2401

9/49=441/2401

−27/343=−189/2401

81/2401=81/2401

5. Sum the numerators while keeping the common denominator.

−1029+441−189+81=−696

Final Answer

(∑_k=1^4)(−3/7)=−696/2401

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