What is the value of
 
160/2*7 + 160/7*12 + 160/12*17 + 160/17*22 + 160/22*27 + 160/27*32
 
Pls show the working

Asked by Gurinder | 20th Nov, 2015, 07:05: PM

Expert Answer:

C o n s i d e r space t h e space g i v e n space e x p r e s s i o n colon fraction numerator 160 over denominator 2 cross times 7 end fraction plus fraction numerator 160 over denominator 7 cross times 12 end fraction plus fraction numerator 160 over denominator 12 cross times 17 end fraction plus fraction numerator 160 over denominator 17 cross times 22 end fraction plus fraction numerator 160 over denominator 22 cross times 27 end fraction plus fraction numerator 160 over denominator 27 cross times 32 end fraction equals 160 cross times open square brackets fraction numerator 1 over denominator 2 cross times 7 end fraction plus fraction numerator 1 over denominator 7 cross times 12 end fraction plus fraction numerator 1 over denominator 12 cross times 17 end fraction plus fraction numerator 1 over denominator 17 cross times 22 end fraction plus fraction numerator 1 over denominator 22 cross times 27 end fraction plus fraction numerator 1 over denominator 27 cross times 32 end fraction close square brackets L e t space S equals 160 cross times open square brackets fraction numerator 1 over denominator 2 cross times 7 end fraction plus fraction numerator 1 over denominator 7 cross times 12 end fraction plus fraction numerator 1 over denominator 12 cross times 17 end fraction plus fraction numerator 1 over denominator 17 cross times 22 end fraction plus fraction numerator 1 over denominator 22 cross times 27 end fraction plus fraction numerator 1 over denominator 27 cross times 32 end fraction close square brackets L e t space u s space f i n d space t h e space g e n e r a l space t e r m space o f space t h e space a b o v e space s e r i e s. T h u s comma space S equals 160 sum from k equals 1 to 6 of fraction numerator 1 over denominator open parentheses 5 k minus 3 close parentheses open parentheses 5 k plus 2 close parentheses end fraction H e r e space sum for blank of i s space t h e space s u m m a t i o n space o f space a l l space t h e space t e r m s. S u b s t i t u t i n g space t h e space v a l u e s space k equals 1 comma 2 comma 3 comma 4 comma 5 space a n d space 6 space w e space h a v e comma S equals 160 cross times open square brackets fraction numerator 1 over denominator 2 cross times 7 end fraction plus fraction numerator 1 over denominator 7 cross times 12 end fraction plus fraction numerator 1 over denominator 12 cross times 17 end fraction plus fraction numerator 1 over denominator 17 cross times 22 end fraction plus fraction numerator 1 over denominator 22 cross times 27 end fraction plus fraction numerator 1 over denominator 27 cross times 32 end fraction close square brackets T h e r e f o r e space w e space n e e d space t o space f i n d comma S equals 160 sum from k equals 1 to 6 of fraction numerator 1 over denominator open parentheses 5 k minus 3 close parentheses open parentheses 5 k plus 2 close parentheses end fraction R e a r r a n g i n g space t h e space t e r m s space i n space n u m e r a t o r comma space w e space h a v e comma S equals 160 sum from k equals 1 to 6 of fraction numerator 5 k plus 2 minus open parentheses 5 k minus 3 close parentheses over denominator 5 open parentheses 5 k minus 3 close parentheses open parentheses 5 k plus 2 close parentheses end fraction rightwards double arrow S equals 160 over 5 sum from k equals 1 to 6 of fraction numerator 5 k plus 2 minus open parentheses 5 k minus 3 close parentheses over denominator open parentheses 5 k minus 3 close parentheses open parentheses 5 k plus 2 close parentheses end fraction rightwards double arrow S equals 32 sum from k equals 1 to 6 of open parentheses fraction numerator 1 over denominator open parentheses 5 k minus 3 close parentheses end fraction minus fraction numerator 1 over denominator open parentheses 5 k plus 2 close parentheses end fraction close parentheses rightwards double arrow S equals 32 open square brackets sum from k equals 1 to 6 of fraction numerator 1 over denominator open parentheses 5 k minus 3 close parentheses end fraction minus sum from k equals 1 to 6 of fraction numerator 1 over denominator open parentheses 5 k plus 2 close parentheses end fraction close square brackets rightwards double arrow S equals 32 open square brackets 1 half minus 1 over 7 plus 1 over 7 minus 1 over 12 plus 1 over 12 minus 1 over 17 plus 1 over 17 minus 1 over 22 plus 1 over 22 minus 1 over 27 plus 1 over 27 minus 1 over 32 close square brackets rightwards double arrow S equals 32 open square brackets 1 half minus 1 over 32 close square brackets rightwards double arrow S equals 32 open square brackets 16 over 32 minus 1 over 32 close square brackets rightwards double arrow S equals 32 open square brackets fraction numerator 16 minus 1 over denominator 32 end fraction close square brackets rightwards double arrow S equals 32 open square brackets 15 over 32 close square brackets rightwards double arrow S equals 15

Answered by Vimala Ramamurthy | 21st Nov, 2015, 09:02: AM