Question

The sum of first n terms of an A.P. is given by \(S_n = 4n^2 - n\). Find the 25th term of this A.P.

• A. 195
• B. 198
• C. 203
• D. 190

Hint:

If \(S_n = An^2 + Bn\), then the \(n\)-th term of the A.P. is given by the general shortcut formula:
\[ a_n = 2An + (B - A) \]
For \(S_n = 4n^2 - n\), we have \(A = 4\) and \(B = -1\).
Substituting these values:
\[ a_n = 2(4)n + (-1 - 4) = 8n - 5 \]
Now, simply substitute \(n = 25\) to get the term:
\[ a_{25} = 8(25) - 5 = 200 - 5 = 195 \]
This shortcut is extremely powerful for quick MCQs!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The sum of the first \(n\) terms of an Arithmetic Progression (A.P.) is represented by the formula \(S_n = 4n^2 - n\). We need to calculate the value of the 25th term of this progression, denoted as \(a_{25}\).

Step 2: Key Formula or Approach:
There are two primary methods to find the \(n\)-th term (\(a_n\)) when the sum formula \(S_n\) is given:

Method 1: Use the relationship between the \(n\)-th term and the sum of terms:
\[ a_n = S_n - S_{n-1} \]

Method 2: Generate the first few terms of the sequence using \(S_n\), determine the first term \(a_1\) and the common difference \(d\), and then use the general term formula:
\[ a_n = a_1 + (n - 1)d \]

Step 3: Detailed Explanation:
Let's solve using

Method 1:
1. The 25th term is given by:
\[ a_{25} = S_{25} - S_{24} \]
2. Calculate the sum of the first 25 terms \(S_{25}\):
\[ S_{25} = 4(25)^2 - 25 \]
\[ S_{25} = 4(625) - 25 \]
\[ S_{25} = 2500 - 25 = 2475 \]
3. Calculate the sum of the first 24 terms \(S_{24}\):
\[ S_{24} = 4(24)^2 - 24 \]
\[ S_{24} = 4(576) - 24 \]
\[ S_{24} = 2304 - 24 = 2280 \]
4. Subtract \(S_{24}\) from \(S_{25}\) to find \(a_{25}\):
\[ a_{25} = S_{25} - S_{24} \]
\[ a_{25} = 2475 - 2280 = 195 \]

Let's verify using

Method 2:
1. Find \(S_1\), which is equal to the first term \(a_1\):
\[ a_1 = S_1 = 4(1)^2 - 1 = 3 \]
2. Find \(S_2\), which is the sum of the first two terms \(a_1 + a_2\):
\[ S_2 = 4(2)^2 - 2 = 16 - 2 = 14 \]
3. Find the second term \(a_2\):
\[ a_2 = S_2 - a_1 = 14 - 3 = 11 \]
4. Calculate the common difference \(d\):
\[ d = a_2 - a_1 = 11 - 3 = 8 \]
5. Calculate the 25th term \(a_{25}\):
\[ a_{25} = a_1 + 24d \]
\[ a_{25} = 3 + 24(8) \]
\[ a_{25} = 3 + 192 = 195 \]
Both methods yield the same result of 195.

Step 4: Final Answer:
The 25th term of the A.P. is 195, which corresponds to option (A).