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Arithmetic Progressions
EXERCISE 5.2
CBSE_NCERT_TEXTBOOK
Grade 10

Question:

Find the 20th term from the last term of the AP : 3, 8, 13, . . ., 253.

Step-by-Step Solution

Key Concept: Use the nth‑term formula of an AP, \(a_n = a + (n-1)d\), to first determine the total number of terms. Then locate the required term by counting backwards from the last term.
1. Identify the first term \(a\) and common difference \(d\):
\[ a = 3, \quad d = 8-3 = 5. \]

2. Let the total number of terms be \(n\). The last term given is \(a_n = 253\). Use the nth‑term formula:
\[ a_n = a + (n-1)d \]
\[ 253 = 3 + (n-1)\times5 \]
\[ (n-1)\times5 = 250 \]
\[ n-1 = 50 \]
\[ n = 51. \]
Hence the AP contains 51 terms.

3. "20th term from the last term" means we count the last term as the 1st term from the end. Therefore the required term is the \( (n-20+1)^{\text{th}} \) term:
\[ \text{Position} = 51 - (20-1) = 32. \]

4. Find the 32nd term using the nth‑term formula:
\[ a_{32} = a + (32-1)d \]
\[ a_{32} = 3 + 31\times5 \]
\[ a_{32} = 3 + 155 = 158. \]

5. Hence the 20th term from the last term of the given AP is 158.

Correct Answer: 158
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