if S subscript n denotes the sum of first n terms of an AP, then prove that S subscript 12 space equals space 3 space left parenthesis space S subscript 8 space minus space S subscript 4 space end subscript right parenthesis space.

Asked by khushalk2903 | 19th Aug, 2019, 09:54: PM

Expert Answer:

begin mathsize 16px style We space know space that comma
straight S subscript straight n space equals space straight n over 2 open square brackets 2 straight a space plus space open parentheses straight n space minus space 1 close parentheses straight d close square brackets
straight L. straight H. straight S space equals space straight S subscript 12 space equals space 12 over 2 open square brackets 2 straight a space plus space open parentheses 12 minus space 1 close parentheses straight d close square brackets space equals space 6 left parenthesis 2 straight a space plus space 11 straight d right parenthesis space equals space 12 straight a space plus space 66 straight d space space space..... space left parenthesis 1 right parenthesis
straight S subscript 8 space equals space 8 over 2 open square brackets 2 straight a space plus space open parentheses 8 minus space 1 close parentheses straight d close square brackets space equals space 4 left parenthesis 2 straight a space plus space 7 straight d right parenthesis space equals space 8 straight a space plus space 28 straight d
straight S subscript 4 space equals space 4 over 2 open square brackets 2 straight a space plus space open parentheses 4 minus space 1 close parentheses straight d close square brackets space equals space 2 left parenthesis 2 straight a space plus space 3 straight d right parenthesis space equals space 4 straight a space plus space 6 straight d

straight R. straight H. straight S space equals space 3 open parentheses straight S subscript 8 space space minus space straight S subscript 4 space close parentheses space equals space 3 open parentheses space 8 straight a space plus space 28 straight d space minus space 4 straight a space minus space 6 straight d close parentheses space equals space 12 straight a space plus space 66 straight d space space space... space left parenthesis 2 right parenthesis

From space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis comma space we space get
straight S subscript 12 space equals space 3 open parentheses straight S subscript 8 space space minus space straight S subscript 4 space close parentheses
end style

Answered by Yasmeen Khan | 20th Aug, 2019, 10:38: AM