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

1 answer

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 \(d\).