If S be the sum of (2n+1) terms of an AP and S1 that of alternate terms beginning with the first ,then show that S/S1 =2n+1/n+1

Asked by nk.pandit02 | 13th Mar, 2017, 07:03: PM

Expert Answer:

begin mathsize 16px style Let space the space AP space be space straight x subscript 1 comma space straight x subscript 2 comma space....... comma space straight x subscript straight n comma space... straight x subscript 2 straight n plus 1 end subscript
and space let space the space first space term space straight x subscript 1 equals straight a space and space the space common space difference space be space straight d.
Recall space that space straight S equals straight n over 2 open square brackets 2 straight a plus left parenthesis straight n minus 1 right parenthesis straight d close square brackets
So comma space straight S equals fraction numerator 2 straight n plus 1 over denominator 2 end fraction open square brackets 2 straight a plus left parenthesis 2 straight n plus 1 minus 1 right parenthesis straight d close square brackets
rightwards double arrow straight S equals fraction numerator 2 straight n plus 1 over denominator 2 end fraction open square brackets 2 straight a plus 2 nd close square brackets
rightwards double arrow straight S equals fraction numerator 2 straight n plus 1 over denominator 2 end fraction open square brackets 2 straight a plus 2 nd close square brackets
rightwards double arrow straight S equals left square bracket 2 straight n plus 1 right square bracket open square brackets straight a plus nd close square brackets space space space..... left parenthesis straight i right parenthesis

The space alternate space terms space would space be space straight x subscript 1 comma space straight x subscript 3 comma space straight x subscript 5 comma..... straight x subscript 2 straight n plus 1 end subscript
So comma space the space number space of space terms space equals straight n plus 1
Here space the space first space term space equals straight a space and space the space common space difference space equals straight a plus 2 straight d minus straight a equals 2 straight d
straight S subscript 1 equals straight a plus left parenthesis straight a plus 2 straight d right parenthesis plus left parenthesis straight a plus 4 straight d right parenthesis plus space. space. space. space. space. space plus left parenthesis straight a plus 2 nd right parenthesis
rightwards double arrow straight S subscript 1 equals left parenthesis straight n plus 1 right parenthesis straight a plus 2 fraction numerator straight n left parenthesis straight n plus 1 right parenthesis straight d over denominator 2 end fraction space space space
space space space space space space space space space space space space space... open square brackets Since space there space space will space be space left parenthesis straight n plus 1 right parenthesis space terms space that space are space apostrophe straight a apostrophe comma space and space you space will space have space 2 straight d plus 4 straight d plus.... plus 2 nd equals 2 straight d left parenthesis 1 plus 2 plus 3 plus.... plus straight n right parenthesis space and space sum space of space 1 plus 2 plus 3 plus.... plus straight n equals fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction. close square brackets
rightwards double arrow straight S subscript 1 equals left parenthesis straight n plus 1 right parenthesis straight a plus straight n left parenthesis straight n plus 1 right parenthesis straight d
rightwards double arrow straight S subscript 1 equals left parenthesis straight n plus 1 right parenthesis left square bracket straight a plus nd right square bracket space space..... left parenthesis ii right parenthesis
Fom space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space get
So comma space straight S over straight S subscript 1 equals fraction numerator left square bracket 2 straight n plus 1 right square bracket open square brackets straight a plus nd close square brackets over denominator left parenthesis straight n plus 1 right parenthesis left square bracket straight a plus nd right square bracket end fraction
rightwards double arrow straight S over straight S subscript 1 equals fraction numerator 2 straight n plus 1 over denominator straight n plus 1 end fraction end style
 
Dear student, you posted this same question twice. Be careful next time since you would have lost your 1 chance to post a question.

Answered by Rebecca Fernandes | 13th Mar, 2017, 08:10: PM

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