What is the sum of the first 21 terms of the series given below?1 – 5 + 4 – 9 + 7 – 13 + 10 – 17 + ……………

1. Break Down the Problem

We need to identify the pattern in the given series and compute the sum of the first 21 terms.

The series is as follows:

• The first term is $1$
• The second term is $-5$
• The third term is $4$
• The fourth term is $-9$
• The fifth term is $7$
• The sixth term is $-13$
• The seventh term is $10$
• The eighth term is $-17$

2. Relevant Concepts

To solve this, we can identify the pattern for the terms:

• Positive terms: $1, 4, 7, 10, \ldots$
• Negative terms: $-5, -9, -13, -17, \ldots$

Positive terms: This is an arithmetic series where:

• First term $a_1 = 1$
• Common difference $d = 3$

The $n$-th positive term can be formulated as: $P_n = 1 + 3(n-1) = 3n - 2$

Negative terms: This is also an arithmetic series where:

• First term $a_1 = -5$
• Common difference $d = -4$

The $n$-th negative term can be formulated as: $N_n = -5 - 4(n-1) = -4n - 1$

3. Analysis and Detail

The sequence of terms alternates between positive and negative. Thus, we can express the total sum of the first 21 terms as follows:

• For the first 21 terms:
  • There are 11 positive terms (since $(21 + 1) / 2 \to 11$)
  • There are 10 negative terms (since $21 / 2 \to 10$)

Sum of positive terms $S_P$: $S_P = P_1 + P_2 + \ldots + P_{11}$ $= (3 \times 1 - 2) + (3 \times 2 - 2) + \ldots + (3 \times 11 - 2)$ $= (1) + (4) + (7) + ... + (31)$ This is the arithmetic series with:

• First term $a = 1$
• Last term $l = 31$
• Number of terms $n = 11$

The sum $S_P$ is calculated as: $S_P = \frac{n}{2} (a + l)$ $= \frac{11}{2} (1 + 31) = \frac{11}{2} \times 32 = 11 \times 16 = 176$

Sum of negative terms $S_N$: $S_N = N_1 + N_2 + \ldots + N_{10}$ $= (-5) + (-9) + (-13) + ... + (-41)$ This is also an arithmetic series with:

• First term $a = -5$
• Last term $l = -41$
• Number of terms $n = 10$

The sum $S_N$ is calculated as: $S_N = \frac{n}{2} (a + l)$ $= \frac{10}{2} (-5 - 41) = 5 \times (-46) = -230$

4. Verify and Summarize

Now we can find the complete sum of the first 21 terms: $\text{Total Sum} = S_P + S_N = 176 - 230 = -54$

Final Answer

The sum of the first 21 terms of the series is $-54$.