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ICSE Class 10 Answered

Given 1/(1*2*3)+1/(2*3*4)+.....+1/(98*99*100)=a/b. Find (b-a).
Asked by arpitadas4117 | 17 Jan, 2022, 02:13: PM
answered-by-expert Expert Answer
fraction numerator 1 over denominator 1 cross times 2 cross times 3 end fraction plus fraction numerator 1 over denominator 2 cross times 3 cross times 4 end fraction plus... plus fraction numerator 1 over denominator 98 cross times 99 cross times 100 end fraction equals straight a over straight b
straight T subscript straight n equals fraction numerator 1 over denominator straight n open parentheses straight n plus 1 close parentheses open parentheses straight n plus 2 close parentheses end fraction equals 1 half open square brackets fraction numerator 1 over denominator straight n open parentheses straight n plus 1 close parentheses end fraction minus fraction numerator 1 over denominator open parentheses straight n plus 1 close parentheses open parentheses straight n plus 2 close parentheses end fraction close square brackets
straight T subscript 1 equals 1 half open square brackets fraction numerator 1 over denominator 1 cross times 2 end fraction minus fraction numerator 1 over denominator 2 cross times 3 end fraction close square brackets
straight T subscript 2 equals 1 half open square brackets fraction numerator 1 over denominator 2 cross times 3 end fraction minus fraction numerator 1 over denominator 3 cross times 4 end fraction close square brackets
straight T subscript 3 equals 1 half open square brackets fraction numerator 1 over denominator 3 cross times 4 end fraction minus fraction numerator 1 over denominator 4 cross times 5 end fraction close square brackets
...
straight T subscript 98 equals 1 half open square brackets fraction numerator 1 over denominator 98 cross times 99 end fraction minus fraction numerator 1 over denominator 99 cross times 100 end fraction close square brackets
fraction numerator 1 over denominator 1 cross times 2 cross times 3 end fraction plus fraction numerator 1 over denominator 2 cross times 3 cross times 4 end fraction plus... plus fraction numerator 1 over denominator 98 cross times 99 cross times 100 end fraction equals 1 half open square brackets fraction numerator 1 over denominator 1 cross times 2 end fraction minus fraction numerator 1 over denominator 99 cross times 100 end fraction close square brackets equals 1 half open square brackets fraction numerator 4950 minus 1 over denominator 9900 end fraction close square brackets equals 4949 over 19800 equals straight a over straight b
Therefore comma space straight b minus straight a equals 14851
Answered by Renu Varma | 18 Jan, 2022, 01:26: PM

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