find the value of 7.8 bar +7.88 bar +7.888 bar + 7.8888 bar + ….. up to 2014 times (interns of non terminating recurring decimal form
The Correct Answer and Explanation is:
To solve the given sum 7.88‾+7.888‾+7.8888‾+7.88888‾+…7.8\overline{8} + 7.88\overline{8} + 7.888\overline{8} + 7.8888\overline{8} + \dots7.88+7.888+7.8888+7.88888+… up to 2014 terms, we need to break it down systematically and find a general formula.
Step 1: Express the recurring decimals in a convenient form
The recurring decimal form for each term is:
- 7.88‾=7+0.88‾7.8\overline{8} = 7 + 0.8\overline{8}7.88=7+0.88
- 7.888‾=7+0.888‾7.88\overline{8} = 7 + 0.88\overline{8}7.888=7+0.888
- 7.8888‾=7+0.8888‾7.888\overline{8} = 7 + 0.888\overline{8}7.8888=7+0.8888
- And so on…
Each term can be represented as:
7+89, 7+8899, 7+888999,…7 + \frac{8}{9}, \; 7 + \frac{88}{99}, \; 7 + \frac{888}{999}, \dots7+98,7+9988,7+999888,…
Step 2: Generalize the nth term
The nth term of the series has the following pattern:
- The decimal part of the nth term is 8…89…9\frac{8 \dots 8}{9 \dots 9}9…98…8, where there are nnn digits in the numerator and nnn digits in the denominator.
Thus, the nth term TnT_nTn can be written as:Tn=7+888…8⏟n digits999…9⏟n digitsT_n = 7 + \frac{\underbrace{888 \dots 8}_{n \text{ digits}}}{\underbrace{999 \dots 9}_{n \text{ digits}}}Tn=7+n digits999…9n digits888…8
This fraction can be rewritten as:Tn=7+10n−29×(10n−1)T_n = 7 + \frac{10^n – 2}{9 \times (10^n – 1)}Tn=7+9×(10n−1)10n−2
Step 3: Summing the terms
Now that we know the general form of each term, we can sum from n=1n = 1n=1 to 2014 terms. The sum SSS of the first 2014 terms is:S=∑n=12014(7+10n−29×(10n−1))S = \sum_{n=1}^{2014} \left( 7 + \frac{10^n – 2}{9 \times (10^n – 1)} \right)S=n=1∑2014(7+9×(10n−1)10n−2)
The sum of constants:∑n=120147=7×2014=14098\sum_{n=1}^{2014} 7 = 7 \times 2014 = 14098n=1∑20147=7×2014=14098
The second part of the sum involves the fractional terms:∑n=1201410n−29×(10n−1)\sum_{n=1}^{2014} \frac{10^n – 2}{9 \times (10^n – 1)}n=1∑20149×(10n−1)10n−2
This sum is more complex to evaluate by hand but can be computed numerically.
Step 4: Approximate solution
For an approximation, you can use a calculator to estimate the fractional part sum. However, since we are summing many terms, the first part of the sum (the constants) dominates the value, and the fractional sums add only a small correction.
Final Answer:
Thus, the approximate value of the sum is 14098+(a small correction from the fractional sums)14098 + \text{(a small correction from the fractional sums)}14098+(a small correction from the fractional sums), which can be computed using more precise methods or numerical tools.
