Question

10th term from the end of the A.P. \(4, 9, 14, \ldots, 254\) will be

• A. 208
• B. 204
• C. 209
• D. 214

Hint:

Remember: the \(k^{th}\) term from the end of an A.P. is given by \(T = l - (k - 1)d\).

Answer 1

The Correct Option is A

Step 1: Identify given data.
First term, \(a = 4\)
Common difference, \(d = 5\)
Last term, \(l = 254\)

Step 2: Find the total number of terms (n).
Using formula: \[ l = a + (n - 1)d \] \[ 254 = 4 + (n - 1)5 \Rightarrow 250 = 5(n - 1) \Rightarrow n - 1 = 50 \Rightarrow n = 51 \]
Step 3: Find the 10th term from the end.
The \(k^{th}\) term from the end of an A.P. is given by \[ T = l - (k - 1)d \] Substitute \(l = 254\), \(k = 10\), \(d = 5\): \[ T = 254 - (10 - 1) \times 5 = 254 - 45 = 209 \]
Step 4: Correction.
However, depending on the indexing, for 10th term from the end, \(T_{n-k+1}\) can also be used: \[ T_{n-k+1} = a + (n - k)d = 4 + (51 - 10) \times 5 = 4 + 205 = 209 \]
Step 5: Conclusion.
The correct 10th term from the end is 209.