If (10)^9+2(11)^1(10)^8+3(11)^2(10)^7+….+10(11)^9 =k(10)^9 then k is equal to [2014] (A) (121)

step 1 Hello students in this question we have given that the ratio of the mass of pb 206 is 0 .80 gram

If (10)^9+2(11)^1(10)^8+3(11)^2(10)^7+….+10(11)^9 =k(10)^9...

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Key Concepts

Geometric Series

A geometric series is a series of the form a + ar + ar^2 + ... where each term is obtained by multiplying the previous term by a constant called the common ratio. The formula for the sum of a finite or infinite geometric series is fundamental in many area of mathematics and is crucial here for managing terms that involve exponential sequences.

Differentiation of Series

Differentiation of a series involves taking the derivative with respect to a parameter (often the common ratio) in a power series. This technique is particularly useful when terms in the series are multiplied by the index (for example, n*r^(n-1)), allowing one to convert the sum into a derivative of a known geometric series sum.

Series Manipulation and Index Transformation

Series manipulation involves techniques such as re-indexing, factoring out common terms, and writing the series in a form that reveals underlying patterns. By appropriately transforming the index and factoring constants, one simplifies the series and aligns it with known formulas, such as those for geometric series and their derivatives, which makes it easier to evaluate complex sums.

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