Given the sequence of terms, AD CG FK JP, the next term is
Solving the Letter Sequence Puzzle: AD CG FK JP
The question asks to find the next term in the given sequence: AD CG FK JP. We need to identify the pattern governing the sequence.
Identifying the Pattern
Let's analyze the sequence by considering the first and second letters separately. We will use the standard alphabetical positions (A=1, B=2, ..., Z=26).
Analyzing First Letters
The first letters of the terms are A, C, F, J.
- A is the $1^{st}$ letter.
- C is the $3^{rd}$ letter. (Difference from A: $3 - 1 = 2$)
- F is the $6^{th}$ letter. (Difference from C: $6 - 3 = 3$)
- J is the $10^{th}$ letter. (Difference from F: $10 - 6 = 4$)
The differences between consecutive first letters are increasing by 1: 2, 3, 4. The next difference should be $4 + 1 = 5$. Therefore, the next first letter will be the $10 + 5 = 15^{th}$ letter of the alphabet, which is O.
Analyzing Second Letters
The second letters of the terms are D, G, K, P.
- D is the $4^{th}$ letter.
- G is the $7^{th}$ letter. (Difference from D: $7 - 4 = 3$)
- K is the $11^{th}$ letter. (Difference from G: $11 - 7 = 4$)
- P is the $16^{th}$ letter. (Difference from K: $16 - 11 = 5$)
The differences between consecutive second letters are also increasing by 1: 3, 4, 5. The next difference should be $5 + 1 = 6$. Therefore, the next second letter will be the $16 + 6 = 22^{nd}$ letter of the alphabet, which is V.
Determining the Next Term
Combining the next first letter (O) and the next second letter (V), the next term in the sequence is OV.
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 ________
