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Permutations And Combinations

Question
CBSEENMA11013103
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Evaluate:

WiredFaculty

Solution
space space space space space sum from straight k equals 1 to 11 of left parenthesis 2 plus 3 to the power of straight k right parenthesis space equals space left parenthesis 2 plus 3 to the power of 1 right parenthesis space plus space left parenthesis 2 plus 3 squared right parenthesis plus........ plus left parenthesis 2 plus 3 to the power of 11 right parenthesis
                     = (2 + 2 +...... to 11 terms) + (3 + 32 +........to 11 terms)
                     space space equals 2 left parenthesis 11 right parenthesis space plus space fraction numerator 3 left parenthesis 3 to the power of 11 minus 1 right parenthesis over denominator 3 minus 1 end fraction space equals space 22 space plus space 3 over 2 left parenthesis 3 to the power of 11 minus 1 right parenthesis
                       = WiredFacultyWiredFaculty

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