Suppose four distinct positive numbers a 1 , a 2 , a 3 , a 4 are in G.P. Let b 1 = a 1 , b 2 = b 1 + a 2 , b 3 = b 2 + a 3 , and b 4 = b 3 + a 4 . STATEMENT-1: The numbers b 1 , b 2 , b 3 , b 4 are neither in A.P. nor in G.P. STATEMENT-2: Th

Question

Suppose four distinct positive numbers $a_{1}, a_{2}, a_{3}, a_{4}$ are in G.P. Let $b_{1} = a_{1}, b_{2} = b_{1} + a_{2}, b_{3} = b_{2} + a_{3},$ and $b_{4} = b_{3} + a_{4} .$

STATEMENT-1: The numbers $b_{1}, b_{2}, b_{3}, b_{4}$ are neither in A.P. nor in G.P.

STATEMENT-2: The numbers $b_{1}, b_{2}, b_{3}, b_{4}$ are in H.P. [2008]

Smart Achievers · Accepted Answer

(3)

$Given: a_{1}, a_{2}, a_{3}, a_{4} are in G.P.$

$Then b_{1}, b_{2}, b_{3}, b_{4} are the numbers a_{1}, a_{1} + a_{2}, a_{1} + a_{2} + a_{3}, a_{1} + a_{2} + a_{3} + a_{4}$

$or a, a + a r, a + a r + a r^{2}, a + a r + a r^{2} + a r^{3}$

Since, above numbers are neither in A.P. nor in G.P. Therefore,

statement 1 is true.

$Also \frac{1}{a}, \frac{1}{a + a r}, \frac{1}{a + a r + a r^{2}}, \frac{1}{a + a r + a r^{2} + a r^{3}} are not in A.P.$

$∴$ $b_{1}, b_{2}, b_{3}, b_{4} are not in H.P.$

$∴$ Statement 2 is false.

Solution

1 STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is a correct explanation for STATEMENT - 1

2 STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is NOT a correct explanation for STATEMENT - 1

3 STATEMENT - 1 is True, STATEMENT - 2 is False

4 STATEMENT - 1 is False, STATEMENT - 2 is True

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