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Question

Find sum of 3+33+333+3333+....... up to n terms.

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Solution

From the given series, take common factor 3

Thus

S=3(1+11+111+1111+...............nthterm)

Or

S=3{1+(1+10)+(1+10+100)+(1+10+100+1000)+...tonterms}

Thus the nth term.

tn=1+10+100+..............+10n1

This is a G.P. where first term is 1 and common ratio is 10 ,

Thus sum of n terms is

tn=1(110n)110tn=(10n1)9

Or

Thus

S=3ni=1tn=3ni=110n19

Hence the sum,

S=310n9319

Or

S=3{10(110n)(110).9(19).n}

Or

S=3{10.(10n1)81n9}

Thus,

S={10.(10n1)27n3}=10n+1109n27



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