Progression (Sequence and Series)Hard
Question
Let S1, S2...... be squares such that for each n ≥ the length of a side of Sn equals the lenght of a disgonal of Sn+1. If the length of a side of S1 is 10 cm, then for which of the following values of n is the area of Sn less than 1 sq. cm ?
Options
A.7
B.8
C.9
D.10
Solution
Let n a denotes the length of side of the square Sn
We are given n a = length of diagonal of n S1+n.
⇒ an = √2 an+1
⇒ an+1 =
This shows that a1, a2, a3 ..... form a GP with common ratio 1/√2.
Therefore, an = a1
⇒ an = 10 (∵ a1 = 10 given)
⇒ an2 = 100
⇒
≤ 1 (∵ an2 ≤ 1 given)
⇒ 100 ≤ 2n-1
This is possible for n ≥ &
So, (b),(c),(d) are the answers.
We are given n a = length of diagonal of n S1+n.
⇒ an = √2 an+1
⇒ an+1 =
This shows that a1, a2, a3 ..... form a GP with common ratio 1/√2.
Therefore, an = a1
⇒ an = 10
(∵ a1 = 10 given)⇒ an2 = 100
⇒
≤ 1 (∵ an2 ≤ 1 given)⇒ 100 ≤ 2n-1
This is possible for n ≥ &
So, (b),(c),(d) are the answers.
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