An AP consists of 50 terms in which the \(3^{rd}\) term is 12 and the last term is 106. Find the \(29^{th}\) term.
(Hint: If 'a' is the first term and 'd' the common difference, then we arrive at the equations a + 2d = 12 and a + 49d = 106. Solve this pair of linear equations for 'a' and 'd'.)

SOLUTION

Note down the given terms and form equations

Third term,
\(a + 2d = 12\) --- (I)
Last(\(50^{th}\)) term,
\(a + 49d = 106\) --- (II)

Subtract (I) from (II)

\((a + 49d) − (a + 2d) = 106 − 12\)
⇒ \(47d = 94\)
⇒ \(d = 2\)

Find \(a\)

From (I),
\( a + 2(2) = 12\)
⇒ \(a + 4 = 12\)
⇒ \(a = 8\)

Find the \(29^{th}\) term

\(\begin{aligned} a_{29} &= a + 28d \\ &= 8 + 28(2) \\ &= 64\end{aligned}\)
So, \(a_{29} = 64\)

🏆

Final Answer : \(a_{29} = 64\)

Concept Note

If two terms of an AP are known, we can form equations to determine the first term(\(a\)) and common difference(\(d\)). After finding \(a\) and \(d\), any required term can be calculated directly.

An AP consists of 50 terms in which the \(3^{rd}\) term is 12 and the last term is 106. Find the \(29^{th}\) term. (Hint: If 'a' is the first term and 'd' the common difference, then we arrive at the equations a + 2d = 12 and a + 49d = 106. Solve this pair of linear equations for 'a' and 'd'.)

SOLUTION

Concept Note

An AP consists of 50 terms in which the \(3^{rd}\) term is 12 and the last term is 106. Find the \(29^{th}\) term.
(Hint: If 'a' is the first term and 'd' the common difference, then we arrive at the equations a + 2d = 12 and a + 49d = 106. Solve this pair of linear equations for 'a' and 'd'.)