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Arithmetic Progressions
EXERCISE 5.2
CBSE_NCERT_TEXTBOOK
Grade 10
Question:
Choose the correct choice in the following and justify : (i) 30th term of the AP: 10, 7, 4, . . . , is (A) 97 (B) 77 (C) –77 (D) – 87 (ii) 11th term of the AP: – 3, 1 2 , 2, . . ., is (A) 28 (B) 22 (C) –38 (D) – 48 1 2
Step-by-Step Solution
Key Concept: Use the nth‑term formula of an arithmetic progression: \(a_n = a + (n-1)d\), where \(a\) is the first term and \(d\) is the common difference.
(i) 1. Identify the first term \(a = 10\). 2. Find the common difference \(d = 7-10 = -3\). 3. Apply the nth‑term formula for \(n = 30\): $$a_{30} = 10 + (30-1)(-3) = 10 + 29(-3) = 10 - 87 = -77.$$ 4. Hence the 30th term is \(-77\). The correct choice is (C) –77.
(ii) 1. The given AP is \(-3,\; -\frac{1}{2},\; 2,\; \dots\). Thus the first term \(a = -3\). 2. Compute the common difference: $$d = \left(-\frac{1}{2}\right) - (-3) = \frac{5}{2}.$$ (The same difference is confirmed by \(2 - (-\frac{1}{2}) = \frac{5}{2}\).) 3. Use the nth‑term formula for \(n = 11\): $$a_{11} = -3 + (11-1)\left(\frac{5}{2}\right) = -3 + 10\times\frac{5}{2} = -3 + 25 = 22.$$ 4. Hence the 11th term is \(22\). The correct choice is (B) 22.
Correct Answer:(i) C, (ii) B
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