Sum of the first 14 terms of an A.P. is 1505 and its first term is 10 . Find its `25^(th)` term.

Options

A. 370
B. 320
C. 380
D. 390

Answer

To solve the problem step by step, we will use the formulas related to Arithmetic Progression (A.P.).

### Step 1: Identify the given values
- The sum of the first 14 terms (S₁₄) = 1505
- The first term (a) = 10
- Number of terms (n) = 14

### Step 2: Use the formula for the sum of the first n terms of an A.P.
The formula for the sum of the first n terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
Where:
- Sₙ = sum of the first n terms
- a = first term
- d = common difference
- n = number of terms

### Step 3: Substitute the known values into the formula
Substituting the known values into the formula:
\[ 1505 = \frac{14}{2} \times (2 \times 10 + (14 - 1)d) \]
This simplifies to:
\[ 1505 = 7 \times (20 + 13d) \]

### Step 4: Simplify the equation
Now, divide both sides by 7:
\[ \frac{1505}{7} = 20 + 13d \]
Calculating the left side:
\[ 215 = 20 + 13d \]

### Step 5: Solve for d
Now, isolate d:
\[ 215 - 20 = 13d \]
\[ 195 = 13d \]
Now, divide both sides by 13:
\[ d = \frac{195}{13} = 15 \]

### Step 6: Find the 25th term
The formula for the nth term (aₙ) of an A.P. is given by:
\[ a_n = a + (n-1)d \]
To find the 25th term (a₂₅):
\[ a_{25} = a + (25 - 1)d \]
Substituting the values:
\[ a_{25} = 10 + 24 \times 15 \]

### Step 7: Calculate the 25th term
Calculating:
\[ a_{25} = 10 + 360 \]
\[ a_{25} = 370 \]

### Final Answer:
The 25th term of the A.P. is **370**.

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