If sum of the `3 ^(rd)` and the `8^(th)` terms of A.P. is 7 and the sum of the `7^(th)` and the `14 ^(th)` terms is -3, find the `10^(th)` term.

Answer

To solve the problem step by step, we will use the properties of Arithmetic Progressions (A.P.).

### Step 1: Define the terms of the A.P.
Let the first term of the A.P. be \( a \) and the common difference be \( d \).

The \( n^{th} \) term of an A.P. is given by:
\[ T_n = a + (n-1)d \]

### Step 2: Write the expressions for the given terms.
From the problem, we need to find:
- The 3rd term, \( T_3 = a + 2d \)
- The 8th term, \( T_8 = a + 7d \)
- The 7th term, \( T_7 = a + 6d \)
- The 14th term, \( T_{14} = a + 13d \)

### Step 3: Set up the equations based on the given information.
According to the problem:
1. The sum of the 3rd and 8th terms is 7:
\[ T_3 + T_8 = (a + 2d) + (a + 7d) = 2a + 9d = 7 \quad \text{(Equation 1)} \]

2. The sum of the 7th and 14th terms is -3:
\[ T_7 + T_{14} = (a + 6d) + (a + 13d) = 2a + 19d = -3 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations.
Now we have two equations:
1. \( 2a + 9d = 7 \) (Equation 1)
2. \( 2a + 19d = -3 \) (Equation 2)

To eliminate \( a \), we can subtract Equation 1 from Equation 2:
\[ (2a + 19d) - (2a + 9d) = -3 - 7 \]
This simplifies to:
\[ 10d = -10 \]
Dividing both sides by 10 gives:
\[ d = -1 \]

### Step 5: Substitute \( d \) back to find \( a \).
Now, substitute \( d = -1 \) back into Equation 1:
\[ 2a + 9(-1) = 7 \]
This simplifies to:
\[ 2a - 9 = 7 \]
Adding 9 to both sides:
\[ 2a = 16 \]
Dividing by 2 gives:
\[ a = 8 \]

### Step 6: Find the 10th term.
Now that we have \( a = 8 \) and \( d = -1 \), we can find the 10th term:
\[ T_{10} = a + (10-1)d = a + 9d \]
Substituting the values:
\[ T_{10} = 8 + 9(-1) = 8 - 9 = -1 \]

### Final Answer:
The 10th term of the A.P. is \( -1 \).

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