Given a1=1 and an=an−1+4 where n≥2 calculate,
lim
First I calculated few terms a_1=1, a_2=5, a_3=9,a_4=13 etc. So
\lim_{n\to \infty }\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+\cdots+\frac{1}{a_na_{n-1}}=\lim_{n\to \infty }\frac{1}{5}+\frac{1}{5\times9}+\cdots+\frac{1}{a_na_{n-1}}
Now I got stuck. How to proceed further? Should I calculate the sum ? Please help.
Answer
HINT:
\dfrac4{a_ma_{m-1}}=\dfrac{a_m-a_{m-1}}{a_ma_{m-1}}=?
a_m=1+4\cdot(m-1)=?
Do you recognize the Telescoping series?
No comments:
Post a Comment