Maths-
General
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Question

2 times C subscript 0 plus 5 times C subscript 1 plus 8 times C subscript 2 plus horizontal ellipsis plus left parenthesis 2 plus 3 n right parenthesis times C subscript n equals

  1. left parenthesis 4 minus 3 n right parenthesis times 2 to the power of n minus 1 end exponent
  2. left parenthesis 4 minus 3 n right parenthesis times 2 to the power of n plus 1 end exponent
  3. left parenthesis 4 plus 3 n right parenthesis times 2 to the power of n minus 1 end exponent
  4. left parenthesis 2 minus 3 n right parenthesis times 2 to the power of n plus 1 end exponent

Hint:

In this question, we have to find sum of 2 times C subscript 0 plus 5 times C subscript 1 plus 8 times C subscript 2 plus horizontal ellipsis plus left parenthesis 2 plus 3 n right parenthesis times C subscript n equals For this we will use,i f space r equals 0 comma space s o comma space space C subscript r equals 0 space a n d space i f space 2 to the power of n equals C presuperscript n subscript 0 plus C presuperscript n subscript 1 plus C presuperscript n subscript 2 plus C presuperscript n subscript 3......... plus C presuperscript n subscript n.

The correct answer is: left parenthesis 4 plus 3 n right parenthesis times 2 to the power of n minus 1 end exponent


    2. C subscript 0 plus 5. C subscript 1 plus 8 C subscript 2 plus........... plus left parenthesis 3 n plus 4 right parenthesis C subscript n
equals sum from r equals 0 to n of left parenthesis 3 n plus 4 right parenthesis space C subscript r
equals sum from r equals 0 to n of 3 n. space C subscript r plus sum from r equals 0 to n of 2. C subscript r space space space space space space space space space space space space space space space open square brackets a s space r equals 0 comma space s o comma space space C subscript r equals 0 close square brackets
equals sum from r equals 1 to n of 3 n. space C presuperscript n minus 1 end presuperscript subscript r minus 1 end subscript plus sum from r equals 0 to n of 2. C presuperscript n minus 1 end presuperscript subscript r minus 1 end subscript space space space space space space space space space space space open square brackets a s space 2 to the power of n equals C presuperscript n subscript 0 plus C presuperscript n subscript 1 plus C presuperscript n subscript 2 plus C presuperscript n subscript 3......... plus C presuperscript n subscript n close square brackets
equals 3 n.2 to the power of n minus 1 end exponent plus 2.2 to the power of n minus 1 end exponent
equals left parenthesis 2 plus 3 n right parenthesis 2 to the power of n minus 1 end exponent space space space space

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