In a series 21, 18, 15............. which term is -81?

Question

Accepted Answer

To determine which term in the series 21, 18, 15, ... is -81, we need to first identify the pattern of the series. This series is an arithmetic sequence where each term decreases by a constant difference.

1. **Identify the common difference:**
   - The first term (a1) is 21.
   - The second term (a2) is 18.
   - The third term (a3) is 15.
   - The common difference (d) can be calculated as:
     $$
     d = a2 - a1 = 18 - 21 = -3
     $$
   - Similarly, $d = a3 - a2 = 15 - 18 = -3$.

2. **General formula for the nth term of an arithmetic sequence:**
   $$
   an = a1 + (n-1) \times d
   $$
   - Here, $an$ is the nth term, $a1$ is the first term, $d$ is the common difference, and $n$ is the term number.

3. **Substitute the known values to find n when $an = -81$:**
   $$
   -81 = 21 + (n-1) \times (-3)
   $$
   $$
   -81 = 21 - 3(n-1)
   $$
   $$
   -81 = 21 - 3n + 3
   $$
   $$
   -81 = 24 - 3n
   $$
   $$
   -81 - 24 = -3n
   $$
   $$
   -105 = -3n
   $$
   $$
   n = \frac{-105}{-3} = 35
   $$

4. **Conclusion:**
   - The term number $n$ that corresponds to -81 is 35. Therefore, Option 1 is correct.

5. **Explanation for other options:**
   - **Option 2 (38):** If $n = 38$, substituting in the formula gives:
     $$
     an = 21 + (38-1) \times (-3) = 21 + 37 \times (-3) = 21 - 111 = -90
     $$
     - This does not equal -81, so Option 2 is incorrect.
   
   - **Option 3 (40):** If $n = 40$, substituting in the formula gives:
     $$
     an = 21 + (40-1) \times (-3) = 21 + 39 \times (-3) = 21 - 117 = -96
     $$
     - This does not equal -81, so Option 3 is incorrect.
   
   - **Option 4 (41):** If $n = 41$, substituting in the formula gives:
     $$
     an = 21 + (41-1) \times (-3) = 21 + 40 \times (-3) = 21 - 120 = -99
     $$
     - This does not equal -81, so Option 4 is incorrect.

Thus, the correct answer is indeed Option 1, as it is the only option that satisfies the condition of the series for the term -81.