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express the following repeated decimals as a rational number

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a) 0.77777...=0.7

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b) 0.757575...=0.75

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-Find the first term and the common ration

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-Find the general term, an for the given sequence

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-Find a5,a10, a20

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-Find sum of first n terms,Sn for the given sequence

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-Find S10 and S20

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a) 3,-4,(16/3,(-64/9),...

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b) (-1/54),(1/81),(-2/243),...

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Step 1:

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The decimal number is $\"\"$ .

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Here the number of repeating terms are $\"\"$ .

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Consider $\"\"$

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Multiply $\"\"$ on each side.

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$\"\"$

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$\"\"$

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Substract $\"\"$ from $\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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The rational number is $\"\"$ .

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$\"\"$

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The decimal number is $\"\"$ .

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Here the number of repeating terms are $\"\"$ .

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Consider $\"\"$

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Multiply $\"\"$ on each side.

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$\"\"$

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$\"\"$

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Substract $\"\"$ from $\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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The rational number is $\"\"$ .

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-Find the first term and the common ration

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-Find the general term, an for the given sequence

\

-Find a5,a10, a20

\

-Find sum of first n terms,Sn for the given sequence

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-Find S10 and S20

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a) 3,-4,(16/3,(-64/9),...

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b) (-1/54),(1/81),(-2/243),...

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Step 1:

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The sequence is $\"\"$ .

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The first term of the sequence is $\"\"$ .

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The common ratio is $\"\"$ .

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$\"\"$ .

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The common ratio is $\"\"$ .

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The sequence is geometric sequence.

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Step 2:

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Formula for $\"\"$ term in geometric sequence is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$

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The general term is $\"\"$ .

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Step 3:

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Find $\"\"$ and $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Step 4:

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The sum of first $\"\"$ terms in a geometric series is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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The sum of first terms is $\"\"$ .

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Find $\"\"$ and $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Solution:

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The first term is $\"\"$ and common ratio is $\"\"$ .

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The general term is $\"\"$ .

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$\"\"$ , $\"\"$ and $\"\"$ .

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The sum of first terms is $\"\"$ .

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\

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Step 1:

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The sequence is $\"\"$ .

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The first term of the sequence is $\"\"$ .

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The common ratio is $\"\"$ .

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$\"\"$ .

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The common ratio is $\"\"$ .

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The sequence is geometric sequence.

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Step 2:

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Formula for $\"\"$ term in geometric sequence is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$

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The general term is $\"\"$ .

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Step 3:

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Find $\"\"$ and $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Step 4:

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The sum of first $\"\"$ terms in a geometric series is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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The sum of first terms is $\"\"$ .

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Find $\"\"$ and $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Solution:

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The first term is $\"\"$ and common ratio is $\"\"$ .

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The general term is $\"\"$ .

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$\"\"$ , $\"\"$ and $\"\"$ .

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The sum of first terms is $\"\"$ .

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$\"\"$ and $\"\"$ .