Identify the Sequence Pattern
Examine the prime factorization of the given terms:
- Term 1: 30 = 2 $\times$ 3 $\times$ 5
- Term 2: 105 = 3 $\times$ 5 $\times$ 7
- Term 3: 385 = 5 $\times$ 7 $\times$ 11
- Term 4: 1001 = 7 $\times$ 11 $\times$ 13
The pattern reveals that each term is the product of three consecutive prime numbers. The sequence of starting primes used is 2, 3, 5, 7.
Calculate the Fifth Term
The next prime number after 7 is 11. Therefore, the fifth term will be the product of the next three consecutive primes, starting with 11.
The primes are 11, 13, and 17.
Calculation:
Fifth Term = 11 $\times$ 13 $\times$ 17
First, calculate 11 $\times$ 13:
$11 \times 13 = 143$
Next, multiply the result by 17:
$143 \times 17 = 2431$
The fifth term of the sequence is 2431.
Important Questions from Series
- In the sequence 6, 9, 14, $x$, 30, 41, a possible value of $x$ is
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 ________
