The greatest integer less than or equal to the sum of first \(100\) terms of the sequence \(\frac 13, \frac 59, \frac {19}{27}, \frac {65}{81},…...\) is equal to ______.
Correct Answer: 98
Solution and Explanation
\(\frac 13, \frac 59, \frac {19}{27}, \frac {65}{81},…...\)
\(S= \displaystyle\sum_{r=1}^{100} (\frac {3^r−2^r}{3^r})\)
\(S=100−\frac 23 \frac {(1−(\frac 23)^{100})}{\frac 13}\)
\(S=98+2(\frac 23)^{100}\)
\(S = 98\)
So, the answer is \(98\).
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Concepts Used:
Principle of Mathematical Induction
The principle of mathematical induction is a powerful technique used to prove that a statement is true for every natural number. The basic idea behind the principle is that if we can show that a statement is true for a base case (usually n=1 or n=0), and if we can show that whether the statement is true for some natural number ‘n’, then it must also be true for the next natural number n+1, then we can conclude that the statement is true for all natural numbers.
To use mathematical induction, we first prove the base case. Then we assume that the statement is true for some arbitrary natural number k, and use this assumption to prove that the statement is also true for k+1. This establishes that the statement is true for all natural numbers greater than or equal to the base case.
The principle of mathematical induction is widely used in mathematics, especially in number theory and combinatorics. It is also used in computer science to prove the correctness of algorithms and data structures.