Question

If 14th term of an A.P. is 4 and its 15th term is zero, then its first term is

• A. –48
• B. –56
• C. 56
• D. 48

Hint:

Notice that the common difference \(d\) is simply the difference between any two consecutive terms:
\[ d = a_{15} - a_{14} = 0 - 4 = -4 \]
Once you have \(d = -4\), you can quickly go backwards from the \(14^{\text{th}}\) term to the \(1^{\text{st}}\) term by adding \(13 \times 4\):
\[ a = a_{14} - 13d = 4 - 13(-4) = 4 + 52 = 56 \]
This reduces calculations to basic arithmetic.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given two specific terms of an Arithmetic Progression (A.P.):
- The \(14^{\text{th}}\) term (\(a_{14}\)) is 4.
- The \(15^{\text{th}}\) term (\(a_{15}\)) is 0.
We need to determine the first term (\(a\)) of this sequence.

Step 2: Key Formula or Approach:
The general term (\(n^{\text{th}}\) term) of an Arithmetic Progression is given by the formula:
\[ a_n = a + (n - 1)d \]
Where:
- \(a\) is the first term.
- \(d\) is the common difference.
We can set up two linear equations with variables \(a\) and \(d\), solve for \(d\), and then substitute it back to find \(a\).

Step 3: Detailed Explanation:

• Write down the expressions for the given terms using the general A.P. formula:
- For the \(14^{\text{th}}\) term (\(n = 14\)):
\[ a_{14} = a + 13d = 4 \] --- (Equation 1)
- For the \(15^{\text{th}}\) term (\(n = 15\)):
\[ a_{15} = a + 14d = 0 \] --- (Equation 2)

• Let us find the common difference \(d\) by subtracting Equation 1 from Equation 2:
\[ (a + 14d) - (a + 13d) = 0 - 4 \] \[ d = -4 \]

• Now, substitute the value \(d = -4\) back into Equation 1 to solve for the first term \(a\):
\[ a + 13(-4) = 4 \] \[ a - 52 = 4 \]

• Isolate the variable \(a\):
\[ a = 4 + 52 \] \[ a = 56 \]


Step 4: Final Answer:
The first term of the A.P. is 56. This corresponds to option (C).