The sum 0.555 …5ndetimus +0.535353 …53=decimal where, n is even, is equal to (a) 12(1-(1)/(10^=

step 1 Hi, we have to work on this N is even here. So, this can write 0 .555 and so on till 5 .2nd deci

The sum 0.555 …5ndetimus +0.535353 …53=decimal where, n is...

The sum $\underbrace{0.555 \ldots 5}_{\text {ndetimus }}+\underbrace{0.535353 \ldots 53}_{=\text {decimal }}$ where, $\mathrm{n}$ is even, is equal to (a) $12\left(1-\frac{1}{10^{=}}\right)$ (b) $\frac{12}{11}\left(1-\frac{1}{10^{-}}\right)$ (c) $\frac{12}{11}\left(1-\frac{1}{10^{2 n}}\right)$ (d) $12\left(1-\left(\frac{1}{10}\right)^{=/ 2}\right)$

The sum $\underbrace{0.555 \ldots 5}_{\text {ndetimus }}+\underbrace{0.535353 \ldots 53}_{=\text {decimal }}$ where, $\mathrm{n}$ is even, is equal to
(a) $12\left(1-\frac{1}{10^{=}}\right)$
(b) $\frac{12}{11}\left(1-\frac{1}{10^{-}}\right)$
(c) $\frac{12}{11}\left(1-\frac{1}{10^{2 n}}\right)$
(d) $12\left(1-\left(\frac{1}{10}\right)^{=/ 2}\right)$

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Key Concepts

Repeating Decimals

A repeating decimal is a decimal number in which one or more digits recur infinitely after the decimal point. It is important in mathematics to identify and convert these decimals into fractional form for exact calculations, especially when performing arithmetic operations.

Conversion of Repeating Decimals to Fractions

The process of expressing a repeating decimal as a fraction involves setting the decimal equal to a variable, multiplying by an appropriate power of 10 to align the repeated digits, and subtracting to eliminate the repeating part. This algebraic manipulation leads to a simple fraction that exactly represents the repeating decimal.

Finite Decimal Representation

Finite decimals have a limited number of digits after the decimal point and can be directly converted into fractions with denominators that are powers of ten. Recognizing finite decimals is essential when summing them with repeating decimals, as each contributes differently to the overall fractional expression.

Infinite Geometric Series

When converting a repeating decimal to a fraction, the repeating part is often represented as an infinite geometric series. Understanding the sum of an infinite geometric series, where each term is a constant multiple of the previous one, is key to deriving the equivalent fraction of the repeating decimal.

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