Sequence Pattern Analysis
The problem asks to find a possible value for '$x$' in the given number sequence: 6, 9, 14, $x$, 30, 41.
We need to identify the pattern governing the sequence.
Identifying the Pattern
Let's examine the differences between consecutive terms:
- $9 - 6 = 3$
- $14 - 9 = 5$
- Let the difference between $x$ and 14 be $d_1$. So, $x - 14 = d_1$.
- Let the difference between 30 and $x$ be $d_2$. So, $30 - x = d_2$.
- $41 - 30 = 11$
The sequence of differences starts with 3, 5, ... , 11.
Calculating the Missing Term '$x$'
Observe the pattern in the differences: 3, 5. It appears the difference increases by 2 each time.
- The difference after 5 should be $5 + 2 = 7$.
- Therefore, $d_1 = 7$.
- Using $x - 14 = d_1$, we get $x - 14 = 7$.
- Solving for $x$: $x = 14 + 7 = 21$.
Let's verify this value using the next difference:
- The difference after $d_1$ (which is 7) should be $7 + 2 = 9$.
- So, $d_2$ should be 9.
- Let's check if $30 - x = 9$ when $x = 21$.
- $30 - 21 = 9$. This matches the expected pattern.
The complete sequence of differences is 3, 5, 7, 9, 11, which follows a consistent pattern of adding 2.
Thus, the possible value of $x$ is 21.
Important Questions from Series
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)
- Calculate the reciprocal of the coefficient of $z^3$ in the Taylor series expansion of the function $f(z) = \sin(z)$ around $z = 0$. (Provide the answer as an integer.)
- Let $a_1 = 1$ and $a_n = a_{n-1} + 4$, $n \ge 2$. Then,
$\lim_{n\to\infty} \left[\frac{1}{a_1a_2} + \frac{1}{a_2a_3} + \dots + \frac{1}{a_{n-1}a_n}\right]$
is equal to ________ - Let $S(x) = a_0 + \sum_{n=1}^\infty(a_n \cos (n x) + b_n \sin (n x))$ be the Fourier series of the$2 \pi$ periodic function defined by $f(x) = x^2 + 4 \sin (x) \cos(x)$, $-\pi \le x \le \pi$. Then
$|\sum_{n=0}^\infty a_n - \sum_{n=1}^\infty b_n|$
is equal to ________ Let $S_n = \sum_{k=1}^n \frac{1}{k}$ and $I_n = \int_1^n \frac{x - [x]}{x^2} dx$. Then, $S_{10} + I_{10}$ is equal to
