Read the given statements carefully: (i) If a,b,care in

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Solution

Q.
Read the given statements carefully:

(i) If a,b,c are in AP then bc, ca, ab is also in AP.

(ii) If each number in an AP is multiplied by a constant number, then the resulting pattern of numbers also forms an AP.

(iii) If each term of an AP is divided by a constant number, then the resulting pattern will form an AP.

(iv) If a constant number is subtracted from each term of an AP, then the resulting pattern of numbers also forms an AP.

Choose the correct option from the following:

1 (i) and (ii)

2 (ii), (iii) and (iv)

3 (i), (ii) and (iii)

4 All are correct

Ans.
(2)

Given, a,b,c are in AP.

We know that if a,b,c are in AP, then $\frac{a}{k}, \frac{b}{k}, \frac{c}{k}$

$a l s o f o r m a n A P w h e r e k \neq 0 . \frac{a}{a b c}, \frac{b}{a b c}, \frac{c}{a b c} a r e i n A P (H e r e, k = a b c) . \Rightarrow \frac{1}{b c}, \frac{1}{a c}, \frac{1}{a b}$

are in AP. Hence, (i) is incorrect.

(ii) Let an AP have first term a and common difference d .
Then, $a, a + d, a + 2 d, a + 3 d, \dots M u l t i p l y i n g e a c h t e r m o f t h e A P b y k, w e g e t a k, a k + d k, a k + 2 d k, a k + 3 d k, \dots N o w, a_{2} - a_{1} = (a k + d k) - (a k) = d k a_{3} - a_{2} = (a k + 2 d k) - (a k + d k) = d k a_{4} - a_{3} = (a k + 3 d k) - (a k + 2 d k) = d k$

Hence, the difference between consecutive terms is constant ( dk ),
so it is an AP.

Similarly, if a constant k is subtracted from each term of an AP or
each term is divided by a non-zero constant k , the new sequence also forms an AP

Therefore, (ii), (iii), and (iv) are correct.

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