An AP consists of 50 terms of which the 3rd term is 12 and the last term is 106. Find the 29th term.
64
Step 1: Recall the nth term formula of an AP.
For an arithmetic progression (AP), the nth term is given by:
\[a_n = a + (n - 1)d\]
where \(a\) is the first term and \(d\) is the common difference.
Step 2: Use the information about the 3rd term.
The 3rd term is 12, so:
\[a_3 = a + 2d = 12\]
This gives our first equation:
\[a + 2d = 12 \quad ...(1)\]
Step 3: Use the information about the 50th (last) term.
The AP has 50 terms and the last term is 106. So the 50th term is:
\[a_{50} = a + 49d = 106\]
This gives our second equation:
\[a + 49d = 106 \quad ...(2)\]
Step 4: Solve the two equations to find \(d\).
Subtract equation (1) from equation (2):
\[(a + 49d) - (a + 2d) = 106 - 12\]
\[a + 49d - a - 2d = 94\]
\[47d = 94\]
So,
\[d = \dfrac{94}{47} = 2\]
Step 5: Find the first term \(a\).
Substitute \(d = 2\) into equation (1):
\[a + 2d = 12\]
\[a + 2(2) = 12\]
\[a + 4 = 12\]
\[a = 12 - 4 = 8\]
Step 6: Find the 29th term.
Use the nth term formula with \(n = 29\):
\[a_{29} = a + (29 - 1)d = a + 28d\]
Substitute \(a = 8\) and \(d = 2\):
\[a_{29} = 8 + 28 \times 2\]
\[a_{29} = 8 + 56 = 64\]
Conclusion: The 29th term of the AP is 64.