QUESTION 2
Easy
Which term of the AP: 21, 18, 15,... is −81? Also, is 0 a term of this AP? Give reasons for your answer.
SOLUTION
1
Identify the first term and common difference
In the AP: 21, 18, 15,...
first term, \(a = 21\)
common difference, \(d = 18 − 21 = −3\)
first term, \(a = 21\)
common difference, \(d = 18 − 21 = −3\)
2
Find which term is −81
The general term is given by the formula:
\(a_n = a + (n − 1)d\)
⇒ \( a_n = 21 + (n − 1)(−21)\)
⇒ \( a_n = 24 − 3n\)
Set \(24 − 3n = −81\)
⇒ \( −3n = −105\)
⇒ \( n = 35\)
Since \(n\) is a positive number, −81 is the \(35^{th}\) term.
\(a_n = a + (n − 1)d\)
⇒ \( a_n = 21 + (n − 1)(−21)\)
⇒ \( a_n = 24 − 3n\)
Set \(24 − 3n = −81\)
⇒ \( −3n = −105\)
⇒ \( n = 35\)
Since \(n\) is a positive number, −81 is the \(35^{th}\) term.
3
Is 0 a term?
Set \(24 − 3n = 0\)
⇒ \(3n = 24\)
⇒ \( n = 8\)
Since \(n\) is a positive number, 0 is a term of the AP. It is the \(8^{th}\) term.
⇒ \(3n = 24\)
⇒ \( n = 8\)
Since \(n\) is a positive number, 0 is a term of the AP. It is the \(8^{th}\) term.
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Final Answer :
−81 is the \(35^{th}\) term of the AP.
Yes, 0 is a term of the AP. It is the \(8^{th}\) term.
Concept Note
The \(n^{th}\) term formula is:
\(a_n = a + (n − 1)d\), where
\(a = \) first term
\(d = \) common difference
\(n = \) term number