Given the series: 701, 349, 173, 85, 41, 19, 8, 2.5

Let 'X' be the nth term and 'Y' be the (n + 1)th term of the series. Determine which of the following statements are correct:

(i) $X = 2Y + 3$

(ii) $Y = 3X + 2$

(iii) The difference between the 1st and 2nd term is twice the difference between the 2nd and 3rd term, and similarly for other consecutive terms.

Question

Given the series: 701, 349, 173, 85, 41, 19, 8, 2.5

Let 'X' be the nth term and 'Y' be the (n + 1)th term of the series. Determine which of the following statements are correct:

(i) $X = 2Y + 3$

(ii) $Y = 3X + 2$

(iii) The difference between the 1st and 2nd term is twice the difference between the 2nd and 3rd term, and similarly for other consecutive terms.

Accepted Answer

**Solution:**

1. Given series: 701, 349, 173, 85, 41, 19, 8, 2.5

2. Let $X$ be the nth term and $Y$ be the (n+1)th term.

3. Check statement (i): $X = 2Y + 3$
   - For $X=701$, $Y=349$:
     $$2 	imes 349 + 3 = 698 + 3 = 701$$ which is true.
   - For $X=349$, $Y=173$:
     $$2 	imes 173 + 3 = 346 + 3 = 349$$ which is true.
   - Similarly for other terms, this holds.

4. Check statement (ii): $Y = 3X + 2$
   - For $X=701$, $Y=349$:
     $$3 	imes 701 + 2 = 2103 + 2 = 2105 
eq 349$$
   - So, statement (ii) is false.

5. Check statement (iii): Difference between 1st and 2nd term is twice the difference between 2nd and 3rd term, and so on.
   - Difference between 1st and 2nd term:
     $$701 - 349 = 352$$
   - Difference between 2nd and 3rd term:
     $$349 - 173 = 176$$
   - Check if $352 = 2 	imes 176$:
     $$352 = 352$$ which is true.
   - Check next differences:
     $$173 - 85 = 88$$
     $$85 - 41 = 44$$
     $$88 = 2 	imes 44$$ true.
   - Similarly for other terms, this pattern holds.

**Answer:** Only (i) and (iii) are correct.

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