Which term of the AP : 21, 18, 15, . . . is – 81? Also, is any term 0? Give reason for your answer.
Step-by-Step Solution
Key Concept: Use the nth‑term formula of an arithmetic progression, \(a_n = a + (n-1)d\), where \(a\) is the first term and \(d\) is the common difference. Solve the resulting linear equation for \(n\) and check whether \(n\) is a positive integer.
1. Identify the first term and common difference:
\[ a = 21, \quad d = 18-21 = -3 \]
2. Write the general term of the AP:
\[ a_n = a + (n-1)d = 21 + (n-1)(-3) \]
3. Find the term that equals \(-81\):
\[ -81 = 21 - 3(n-1) \]
\[ -81 - 21 = -3(n-1) \]
\[ -102 = -3(n-1) \]
Divide both sides by \(-3\):
\[ 34 = n-1 \]
\[ n = 35 \]
Hence, the 35th term of the AP is \(-81\).
4. Check whether any term is \(0\):
Set \(a_n = 0\):
\[ 0 = 21 - 3(n-1) \]
\[ -3(n-1) = -21 \]
Divide by \(-3\):
\[ n-1 = 7 \]
\[ n = 8 \]
Since \(n = 8\) is a positive integer, the 8th term of the AP is \(0\).
5. Reason: The common difference is \(-3\); each successive term reduces by 3. After 7 steps from the first term (21), the value becomes \(21 - 7\times3 = 0\). Thus a zero term indeed occurs.
6. Answer summary:
- The term that equals \(-81\) is the 35th term.
- Yes, a term equals \(0\); it is the 8th term.
Correct Answer: The 35th term of the AP is \(-81\). A term equal to 0 does occur – it is the 8th term (since \(a_8 = 0\)).