Sequence with ratio

Calculus Level 1

Sequence { a n } \{a_n\} satisfies a 1 = 3 , a n + 1 = a n a n + 1 , a_1=3, a_{n+1}=\frac{a_n}{a_n+1}, where n n is a positive integer. What is the value of a 14 ? a_{14}?

1 20 \frac{1}{20} 1 10 \frac{1}{10} 1 40 \frac{1}{40} 3 40 \frac{3}{40}

Mahindra Jain by Mahindra Jain

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3 solutions

Ercole Suppa
May 3, 2014

Solution. \textbf{Solution.} We have 1 a n + 1 = 1 + 1 a n \frac{1}{a_{n+1}}=1+\frac{1}{a_n} Hence, by putting x n = 1 a n x_n=\dfrac{1}{a_n} , we obtain x n + 1 = x n + 1 x_{n+1}=x_n+1 Since x 1 = 1 3 x_1=\frac{1}{3} it follows x 14 = x 1 + 13 = 1 3 + 13 = 40 30 ⇒ a 14 = 3 40 x_{14}=x_1+13=\frac{1}{3}+13=\frac{40}{30} \qquad \Rightarrow \qquad a_{14}=\frac{3}{40} and we are done.

4 Helpful 0 Interesting 1 Brilliant 0 Confused

hey please explain the first step

Rishabh Jain - 6 years, 11 months ago
Vidya Shirur
Mar 22, 2014

a1=3,a2=3/4,a3=3/7,.................a14=3/40

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Souvik Ghosh
Mar 8, 2014

a1 = 3, so a13= 13*3 =39. a13/(a13+1)= a13/40. a13 can't be 1. gussed 3.

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