How such A.P subsets are there???

[swastik pramanik]

Let \(S=\{1, 2,3,\cdots, n\}\).  Find the number of subsets \(A\) of \(S\) satisfying the following conditions:

• \(A=\{a, a+d, \cdots, a+kd\}\) for some positive integers \(a, d, k\), and
• \(A\cup \{x\} \) is no longer an A.P with common difference \(d\) for each \(x\in S\backslash A\).

(Note that \(|A|\geq 2\)  and any sequence of two terms is considered as an A.P).

[Camellia Ray]

In the given question it has been asked to find the no of subsets which satisfy the condition that
A={a, a+d,a+2d….a+kd} &

A U{x}  is no longer an A.P where x ∈ (S-A)

From above 2 condns it can b concluded that at a time with c.d d we have to form the largest sequence having c.d d and starting element say a

For ex if X={1,2,3,,,,12} then for

c.d 1:  only 1 subset possible; c.d 2 ->2 :subset possible namely { 1,3,5,7,,,11} & {2,4,6,8,,,,12}

Now c.d 1 and c.d 11 only 1 subset; c.d 2 &10 ->2 subsets

GENERALIZATION

For N Odd

1. c.d 1 & n-1 -> 1 subset;  2. c.d 2 & n-2 -> 2 subsets

similarly for c.d (n-1)/2) & cd (n+1)/(2) it will be (n-1)/2 subsets

SO TOTAL IS 2× (1+2+3+…..+(n-1)/2)

FOR EVEN

SAME as previous just one more term n/2 added

so here it is 2×(1+2+….(n−1)/2)+n/2

[Author]

Posts