Find the missing sequence in the letter series.
B, FH, LNP, _ _ _ _.
Letter Series Analysis
The given letter series is B, FH, LNP, _ _ _ _.
We need to identify the pattern to find the missing four-letter sequence.
Pattern Identification
Let's analyze the series based on the number of letters and their positions in the alphabet:
- Term 1: B (1 letter). Position: 2.
- Term 2: FH (2 letters). Positions: 6, 8.
- Term 3: LNP (3 letters). Positions: 12, 14, 16.
1. Number of Letters Pattern
The number of letters in each term follows a simple progression:
- Term 1: 1 letter
- Term 2: 2 letters
- Term 3: 3 letters
- Based on this, Term 4 should have 4 letters.
2. Starting Letter Position Pattern
Let's examine the alphabetical positions of the first letter of each term:
- Term 1 starts with B (position 2).
- Term 2 starts with F (position 6).
- Term 3 starts with L (position 12).
The difference between the starting positions increases:
- Position(F) - Position(B) = 6 - 2 = 4
- Position(L) - Position(F) = 12 - 6 = 6
The difference increases by 2 each time (4, 6). Therefore, the next difference should be $6 + 2 = 8$.
The starting position for Term 4 is Position(L) + 8 = 12 + 8 = 20.
The 20th letter of the alphabet is T. So, Term 4 starts with T.
3. Internal Letter Spacing Pattern
Let's check the gap between consecutive letters within each term (where applicable):
- Term 2 (FH): Position(H) - Position(F) = 8 - 6 = 2.
- Term 3 (LNP): Position(N) - Position(L) = 14 - 12 = 2. Position(P) - Position(N) = 16 - 14 = 2.
The pattern is that consecutive letters within a term are separated by 2 positions.
4. Deriving the Missing Sequence (Term 4)
Term 4 must have 4 letters, start with T (position 20), and maintain the internal gap of 2.
- 1st letter: T (position 20)
- 2nd letter: Position 20 + 2 = 22. The 22nd letter is V.
- 3rd letter: Position 22 + 2 = 24. The 24th letter is X.
- 4th letter: Position 24 + 2 = 26. The 26th letter is Z.
Therefore, the missing sequence is TVXZ.
Conclusion
The pattern involves an increasing number of letters per term, starting positions incrementing by progressively larger even numbers ($+4, +6, +8, ...$), and a constant internal gap of 2 between letters within a term.
The missing sequence is TVXZ.
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 ________
