Question

In an A.P., \(15^{th}\) term exceeds the \(8^{th}\) term by 21. If sum of first 10 terms is 55, then form the A.P.

Hint:

The difference between any two terms \(a_{p}\) and \(a_{q}\) is simply \((p - q)d\). Here, \(a_{15} - a_{8} = 7d\), which allows you to find \(d\) instantly.

Answer 1

Step 1: Understanding the Concept:
An Arithmetic Progression (A.P.) is defined by its first term (\(a\)) and common difference (\(d\)). We use the general term and sum formulas to establish equations.
Step 2: Key Formula or Approach:
\(n^{th}\) term: \(a_{n} = a + (n - 1)d\)
Sum of \(n\) terms: \(S_{n} = \frac{n}{2}[2a + (n - 1)d]\)
Step 3: Detailed Explanation:
Given: \(a_{15} - a_{8} = 21\)
\[ (a + 14d) - (a + 7d) = 21 \]
\[ 7d = 21 \implies d = 3 \]
Given: \(S_{10} = 55\)
\[ \frac{10}{2}[2a + (10 - 1)d] = 55 \]
\[ 5[2a + 9(3)] = 55 \]
\[ 2a + 27 = 11 \]
\[ 2a = 11 - 27 = -16 \]
\[ a = -8 \]
The A.P. is:
Term 1: \(a = -8\)
Term 2: \(a + d = -8 + 3 = -5\)
Term 3: \(a + 2d = -8 + 6 = -2\)
Term 4: \(a + 3d = -8 + 9 = 1\)
Step 4: Final Answer:
The A.P. is \(-8, -5, -2, 1, \dots\)