Sum Of First N Terms Of An Ap

Sum of the n terms of the series

%5Cfrac%7B1%7D%7B2%7D%20%2B%20%5Cfrac%7B3%7D%7B4%7D%20%2B%20%5Cfrac%7B7%7D%7B8%7D%20%2B%20%5Cfrac%7B15%7D%7B16%7D%20%2B%20...

upto n terms can be written as

%5Cbigg(1-%5Cfrac%7B1%7D%7B2%7D%5Cbigg)%20%2B%20%5Cbigg(1-%5Cfrac%7B1%7D%7B4%7D%5Cbigg)%20%2B%20%5Cbigg(1-%5Cfrac%7B1%7D%7B8%7D%5Cbigg)%20%2B%20%5Cbigg(1-%5Cfrac%7B1%7D%7B16%7D%5Cbigg)%20%2B%20...%20%5C%2C%20upto%5C%2C%20n%5C%2C%20terms

%3D%20n%20-%5Cbigg(%5Cfrac%7B1%7D%7B2%7D%20%2B%20%5Cfrac%7B1%7D%7B4%7D%20%2B%20%5Cfrac%7B1%7D%7B8%7D%20%2B%20...%20%2B%20n%5C%2C%20terms%20%5Cbigg)

n%20-%5Cfrac%7B%5Cfrac%7B1%7D%7B2%7D%5Cbigg(1-%5Cfrac%7B1%7D%7B2%5En%7D%20%5Cbigg)%20%7D%7B1-%5Cfrac%7B1%7D%7B2%7D%7D%20%3D%20n%20%2B%202%5E%7B-n%7D-1

Multiplicative loop is very important approach in IIT
Mathematics

%5CBigg(tan%5Cfrac%7B%5Ctheta%7D%7B2%7D%5CBigg)(1%2Bsec%20%5Ctheta)%3D%5Cfrac%7Bsin%20%5Ctheta%2F2%7D%7Bcos%20%5Ctheta%2F2%7D%5CBigg%5B1%2B%5Cfrac%7B1%7D%7Bcos%20%5Ctheta%7D%5CBigg%5D

%3D%5Cfrac%7B(sin%20%5Ctheta%2F2)2%20cos%5E2%20%5Ctheta%2F2%7D%7B(cos%20%5Ctheta%2F2)2%20cos%20%5Ctheta%7D

%3D%5Cfrac%7B(2%20sin%20%5Ctheta%2F2)cos%20%5Ctheta%2F2%7D%7Bcos%20%5Ctheta%7D%3D%5Cfrac%7Bsin%20%5Ctheta%7D%7Bcos%20%5Ctheta%7D%3Dtan%20theta

%5Ctherefore%20f_n(%5Ctheta)%3D(tan%20%5Ctheta%2F2)(1%2Bsec%5Ctheta)

%3D(1%2Bsec%202%5Ctheta)(1%2Bsec2%5E2%5Ctheta)...(1%2Bsec%202%5En%20%5Ctheta)

%3D(tan%20%5Ctheta)(1%2Bsec%202%20%5Ctheta)(1%2Bsec%202%5E2%20%5Ctheta)(1%2Bsec%202%5E2%20%5Ctheta)....(1%2Bsec%202%5En%20%5Ctheta)

tan%202%5Ctheta.(1%2Bsec2%5E2%20%5Ctheta)...(1%2Bsec%202%5En%20%5Ctheta)

%3Dtan(2%5En%20%5Ctheta)

Now,

f_2%5CBigg(%5Cfrac%7B%5Cpi%7D%7B16%7D%5CBigg)%3Dtan%5CBigg(2%5E2%5Cfrac%7B%5Cpi%7D%7B16%7D%5CBigg)%3Dtan%5CBigg(%5Cfrac%7B%5Cpi%7D%7B4%7D%5CBigg)%3D1

Therefore, (a) is the answer.

f_3%5CBigg(%5Cfrac%7B%5Cpi%7D%7B32%7D%5CBigg)%3Dtan%5CBigg(2%5E3%5Cfrac%7B%5Cpi%7D%7B32%7D%5CBigg)%3Dtan%5CBigg(%5Cfrac%7B%5Cpi%7D%7B4%7D%5CBigg)%3D1

Therefore, (b) is the answer.

f_4%5CBigg(%5Cfrac%7B%5Cpi%7D%7B64%7D%5CBigg)%3Dtan%5CBigg(2%5E4%5Cfrac%7B%5Cpi%7D%7B64%7D%5CBigg)%3Dtan%5CBigg(%5Cfrac%7B%5Cpi%7D%7B4%7D%5CBigg)%3D1

Therefore, (c) is the answer.

f_5%5CBigg(%5Cfrac%7B%5Cpi%7D%7B128%7D%5CBigg)%3Dtan%5CBigg(2%5E5%5Cfrac%7B%5Cpi%7D%7B128%7D%5CBigg)%3Dtan%5CBigg(%5Cfrac%7B%5Cpi%7D%7B4%7D%5CBigg)%3D1

Therefore, (d) is the answer.

Chapter Questions
2 MCQs