To find the sum of the series $4+44+444+....$ up to n terms, we first rewrite the terms:
- The k-th term, $T_k$, can be expressed as $4 \times (111....1)$ (k times).
- The number $111....1$ (k times) can be written as $\frac{10^k - 1}{9}$.
- Therefore, the k-th term is $T_k = 4 \times \frac{10^k - 1}{9}$.
Deriving the Sum Formula
The sum of the first n terms, $S_n$, is given by:
$S_n = \sum_{k=1}^{n} T_k = \sum_{k=1}^{n} \left( 4 \times \frac{10^k - 1}{9} \right)$Factor out the constant term:
$S_n = \frac{4}{9} \sum_{k=1}^{n} (10^k - 1)$Separate the summation:
$S_n = \frac{4}{9} \left( \sum_{k=1}^{n} 10^k - \sum_{k=1}^{n} 1 \right)$Evaluating the Summations
- The sum $\sum_{k=1}^{n} 1 = n$.
- The sum $\sum_{k=1}^{n} 10^k$ is a geometric series $10 + 10^2 + ... + 10^n$.
- First term $a = 10$.
- Common ratio $r = 10$.
- Number of terms = $n$.
- The sum is $a \frac{r^n - 1}{r-1} = 10 \frac{10^n - 1}{10-1} = 10 \frac{10^n - 1}{9} = \frac{10^{n+1} - 10}{9}$.
Final Sum Calculation
Substitute the evaluated sums back into the equation for $S_n$:
$S_n = \frac{4}{9} \left( \frac{10^{n+1} - 10}{9} - n \right)$Combine the terms inside the parenthesis over a common denominator:
$S_n = \frac{4}{9} \left( \frac{10^{n+1} - 10 - 9n}{9} \right)$Simplify the expression:
$S_n = \frac{4}{81} (10^{n+1} - 9n - 10)$This result matches the formula in option 3.
Important Questions from Series
The following figures show three curves generated using an iterative algorithm. The total length of the curve generated after 'Iteration n' is:
Note: The figures shown are representative.
Let $a_0 = 0$ and define $a_n = \frac{1}{2}(1 + a_{n-1})$ for all positive integers $n \ge 1$.
The least value of $n$ for which $|1 - a_n| < \frac{1}{2^{10}}$ is __________.
(Answer in integer)
- In the sequence 6, 9, 14, $x$, 30, 41, a possible value of $x$ is
The sum of the first $n$ terms in the sequence 8, 88, 888, 8888, ... is______.
- The difference between the sum of the first $2n$ natural numbers and the sum of the first $n$ odd natural numbers is ______
