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95. Find the sum of the following series up to n terms:

(i) 5 + 55 +555 + …

(ii) .6 +. 66 +. 666+…

0 2 Views | Posted 6 months ago
Asked by Shiksha User

  • 1 Answer

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    Answered by

    Payal Gupta | Contributor-Level 10

    6 months ago

    95. (i) Sn = 5+55+555+…………… upto n term

    =5(1+11+111+ …………….upto n term )

    Multiplying and dividing by 9 we get

    =59(9+99+999+…upto nterms)

    =59[(10−1)+(100−1)+(000−1)+….upto nterms] =59[(10+100+1000+?n)−(1+1+1...upto nterms]

    =59[(10+102+103+?upto nterms
    −n×1]

    =59[10(10n−1)10−1−n] { ?10+102+103+…ntermsisaG.Pofa=10,r=10>1nterms }

    =59×[109(10n−1)−n]

    =5081(10n−1)−5n9

     

    (ii) Sn=0.6+0.66+0.666+…n

    =6[0.1+0.11+0.111+?uptonterms]

    =69[0.9+0⋅99+0.999+?uptonterms] {multiplying and dividing by 9}

    =69[(1−0.1)+(1−0.01)+(1−0.001)+?uptonterms]

    =69[(1+1+1…n)−(0.1+0.01+0.001+…uptonterms)]

    =69[n×1−(110+1100+11000+?uptonterms)]

    =69[n−(110+110×110+110×(110)2+?uptonterms)]

    =69[n−110(1−(110)n)1−110]

    =69[n−1−(110)n10−1]

    =69[x−19(1−110n)]

    =69×19[9x−(1−110n)]

    =227[9n−(1−110n)]

    =227[9n−1+110n]

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