Let Sn(1 ≤n ≤9) denotes the sum of n terms of series 1+22+333+…+999999999, then for 2 ≤n ≤9 (A)

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Let Sn(1 ≤n ≤9) denotes the sum of n terms of series...

Let $S_{n}(1 \leq n \leq 9)$ denotes the sum of $n$ terms of series $1+22+333+\ldots+999999999$, then for $2 \leq n \leq 9$ (A) $S_{n}-S_{n-1}=\frac{1}{9}\left(10^{n}-n^{2}+n\right)$ (B) $S_{n}=\frac{1}{9}\left(10^{n}-n^{2}+2 n-2\right)$ (C) $9\left(S_{n}-S_{n-1}\right)=n\left(10^{n}-1\right)$ (D) None of these

Let $S_{n}(1 \leq n \leq 9)$ denotes the sum of $n$ terms of series $1+22+333+\ldots+999999999$, then for $2 \leq n \leq 9$
(A) $S_{n}-S_{n-1}=\frac{1}{9}\left(10^{n}-n^{2}+n\right)$
(B) $S_{n}=\frac{1}{9}\left(10^{n}-n^{2}+2 n-2\right)$
(C) $9\left(S_{n}-S_{n-1}\right)=n\left(10^{n}-1\right)$
(D) None of these

Key Concepts

Closed-Form Expressions

A closed-form expression represents a sum or series in a finite, non-iterative formula. In problems involving series, having a closed-form expression allows one to compute terms directly without summing sequentially. This approach is fundamental to validating and comparing different proposed formulas for the sum of a series.

Number Representation Using Repeated Digits

Numbers composed of repeated digits, such as 333 or 4444, can be expressed in an algebraic form by recognizing that they consist of multiples of sums of powers of 10. Specifically, a number with n repeated digits d can be written as d × (10^n - 1)/9. This conversion of a numeral pattern into a formula is essential for analyzing and summing series where each term has a repeated-digit structure.

Geometric Series

The sum (10^n - 1)/9 arises from the geometric series 1 + 10 + 10^2 + ... + 10^(n-1). A geometric series is one in which each term is obtained by multiplying the previous term by a constant ratio (in this case, 10). This principle underlies the closed-form expression of numbers with repeating digits and is pivotal in deriving formulas for series sums.

Partial Sums and Term Differences

When dealing with series, it is often useful to consider partial sums and their differences. The relationship S? - S??? represents the nth term of the series. Recognizing this difference is a key step in verifying formulas and understanding the incremental structure of the series, as well as in deriving recurrence relations or closed-form expressions for the sum.

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