In an A.P. the sum of the first ten terms is 210 and thee difference between the first and the last term si 36 . Fiind the first term in the A.P.

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the given information
We are given two pieces of information about an arithmetic progression (A.P.):
1. The sum of the first 10 terms (S₁₀) is 210.
2. The difference between the first term (a) and the last term (l) is 36.

### Step 2: Use the formula for the sum of the first n terms
The formula for the sum of the first n terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (a + l) \]
For our case, where n = 10:
\[ S_{10} = \frac{10}{2} \times (a + l) \]
This simplifies to:
\[ S_{10} = 5 \times (a + l) \]

### Step 3: Substitute the known sum into the equation
We know that \( S_{10} = 210 \), so we can set up the equation:
\[ 5 \times (a + l) = 210 \]
Dividing both sides by 5 gives:
\[ a + l = 42 \]
Let's label this as Equation (1).

### Step 4: Set up the second equation using the difference between terms
We are also given that the difference between the last term and the first term is 36:
\[ l - a = 36 \]
Let's label this as Equation (2).

### Step 5: Solve the system of equations
Now we have two equations:
1. \( a + l = 42 \) (Equation 1)
2. \( l - a = 36 \) (Equation 2)

We can add these two equations together:
\[ (a + l) + (l - a) = 42 + 36 \]
This simplifies to:
\[ 2l = 78 \]
Dividing both sides by 2 gives:
\[ l = 39 \]

### Step 6: Substitute the value of l back into Equation (1)
Now that we have \( l = 39 \), we can substitute this value back into Equation (1):
\[ a + 39 = 42 \]
Subtracting 39 from both sides gives:
\[ a = 42 - 39 \]
Thus:
\[ a = 3 \]

### Step 7: Conclusion
The first term of the A.P. is \( a = 3 \).

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