The sum of 5 – 5 + 5 – 5 + 5 – 5 ……, to odd number of terms is :

Correct Answer: 5

Step-by-step Explanation:

We are given the alternating series:

$5-5+5-5+5-5+\dots$

We need to find the sum of this series up to an odd number of terms.

Let us observe the pattern of partial sums by evaluating the sum for different numbers of terms:

1. For 1 term (odd):

$S_1=5$

2. For 2 terms (even):

$S_2=5-5=0$

3. For 3 terms (odd):

$S_3=5-5+5=5$

4. For 4 terms (even):

$S_4=5-5+5-5=0$

5. For 5 terms (odd):

$S_5=5-5+5-5+5=5$

From the above pattern, we can see that:

- The sum up to any even number of terms is always 0, because every positive term $+5$ is canceled out by a succeeding negative term $-5$.

- The sum up to any odd number of terms is always 5, because all pairs cancel out, leaving just the first single term $5$.

Thus, the sum of the series to an odd number of terms is equal to 5.