Prove that (33 …3)^2n digin +22 …2=figta =111 ….12 n . Hence find the value of 9(11111)^2+2(1

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Prove that (33 …3)^2n digin +22 …2=figta =111 ….12 n . ...

Prove that $$
\underbrace{(33 \ldots 3)^{2}}_{n \text { digin }}+\underbrace{22 \ldots 2}_{=\text {figta }}=\underbrace{111 \ldots .1}_{2 n} \text { . }
$$ Hence find the value of $9(11111)^{2}+2(11111)$.

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Key Concepts

Repunits and Their Properties

Repunits are numbers entirely composed of the digit 1, such as 11, 111, etc. They are usually expressed in terms of powers of 10, for example as (10^n - 1)/9. Understanding repunits is essential for problems where a relationship between numbers with other repeated digits and repunits is established, as it often involves constructing or decomposing numbers into such forms.

Algebraic Manipulation and Pattern Recognition

The core of proving identities involving numbers made up of repeated digits is adept algebraic manipulation. This involves rewriting numbers using their geometric series forms, performing operations like squaring and addition, and then simplifying the expressions to reveal underlying patterns or identities. Recognizing these patterns and applying proper algebraic techniques is crucial to validate the stated identities in such problems.

Geometric Series Representation

Many numerical problems involving repeated digits rely on expressing such numbers as sums of geometric series. For instance, a number consisting entirely of the same digit can be written as that digit multiplied by the sum of successive powers of 10. This approach simplifies handling these numbers algebraically, allowing one to derive closed?form expressions which are pivotal in proofs and simplifications.

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