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
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 ______
