Question

The 10 th and the 15 terms of ap are _5 and _7½ respectively

Answer 1

To find the terms of the arithmetic progression (AP) given that the 10th term is \(-5\) and the 15th term is \(-7.5\), we can use the formula for the \(n\)th term of an AP, which is given by:

\[
a_n = a + (n - 1)d
\]

Where:
- \(a\) is the first term,
- \(d\) is the common difference,
- \(n\) is the term number.

Given the information:
1. For the 10th term:
\[
a + 9d = -5 \quad \text{(1)}
\]

2. For the 15th term:
\[
a + 14d = -7.5 \quad \text{(2)}
\]

Now we can set up the equations based on the above two expressions.

### Step 1: Solve the equations

From equation (1):
\[
a + 9d = -5
\]

From equation (2):
\[
a + 14d = -7.5
\]

### Step 2: Subtract equation (1) from equation (2)
\[
(a + 14d) - (a + 9d) = -7.5 - (-5)
\]
\[
14d - 9d = -7.5 + 5
\]
\[
5d = -2.5
\]
\[
d = -\frac{2.5}{5} = -0.5
\]

### Step 3: Substitute \(d\) back into one of the equations to find \(a\)
Substituting \(d = -0.5\) into equation (1):
\[
a + 9(-0.5) = -5
\]
\[
a - 4.5 = -5
\]
\[
a = -5 + 4.5 = -0.5
\]

### Conclusion
We find:
- \(a = -0.5\)
- \(d = -0.5\)

### Resulting Terms
The general formula for the \(n\)th term becomes:
\[
a_n = -0.5 + (n-1)(-0.5)
\]

Calculating the first few terms:
- 1st term (\(a_1\)):
\[
a_1 = -0.5 + 0 = -0.5
\]
- 2nd term (\(a_2\)):
\[
a_2 = -0.5 - 0.5 = -1
\]
- 3rd term (\(a_3\)):
\[
a_3 = -0.5 - 1 = -1.5
\]

Thus, the AP starts as:
\[
-0.5, -1, -1.5, -2, \ldots
\]

So the 10th and 15th terms are indeed \(-5\) and \(-7.5\), confirming our values for \(a\) and \(d\).