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Home / Doubts and Solutions/KERALA/Class 11 Science/Mathematics

P(n): 1+3+3^2+.....+3^n-1=(3^n-1)/2.
If P(k) is true prove P(k+1) is true. 

Asked by Deebu 1st February 2015, 4:08 PM
Answered by Expert
Answer:
P left parenthesis n right parenthesis colon 1 plus 3 plus 3 squared plus 3 cubed......... plus 3 to the power of n minus 1 end exponent I t space i s space g i v e n space t h a t space P left parenthesis k right parenthesis space i s space t r u e therefore P left parenthesis k right parenthesis colon 1 plus 3 plus 3 squared plus 3 cubed......... plus 3 to the power of k minus 1 end exponent equals fraction numerator 3 to the power of k minus 1 over denominator 2 end fraction...... left parenthesis i right parenthesis F o r space n equals k plus 1 comma space w e space h a v e space t o space p r o v e space t h a t space P left parenthesis k plus 1 right parenthesis colon 1 plus 3 plus 3 squared plus 3 cubed......... plus 3 to the power of k minus 1 end exponent plus 3 to the power of k equals fraction numerator 3 to the power of k plus 1 end exponent minus 1 over denominator 2 end fraction..... left parenthesis i i right parenthesis A d d i n g space 3 to the power of k o n space b o t h space s i d e s space o f space left parenthesis i right parenthesis comma space w e space g e t P left parenthesis k right parenthesis colon 1 plus 3 plus 3 squared plus 3 cubed......... plus 3 to the power of k minus 1 end exponent plus 3 to the power of k equals fraction numerator 3 to the power of k minus 1 over denominator 2 end fraction plus 3 to the power of k equals fraction numerator 3 to the power of k minus 1 plus 2.3 to the power of k over denominator 2 end fraction equals fraction numerator 3 to the power of k plus 2.3 to the power of k minus 1 over denominator 2 end fraction equals fraction numerator 3 to the power of k left parenthesis 1 plus 2 right parenthesis minus 1 over denominator 2 end fraction equals fraction numerator 3 to the power of k.3 minus 1 over denominator 2 end fraction equals fraction numerator 3 to the power of k plus 1 end exponent minus 1 over denominator 2 end fraction therefore P left parenthesis k plus 1 right parenthesis colon 1 plus 3 plus 3 squared plus 3 cubed......... plus 3 to the power of k minus 1 end exponent plus 3 to the power of k equals fraction numerator 3 to the power of k plus 1 end exponent minus 1 over denominator 2 end fraction comma space w h i c h space i s space t h e space s a m e space a s space left parenthesis i i right parenthesis H e n c e space P r o v e d
Answered by Expert 1st February 2015, 7:19 PM
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