iran-konkur 2021 Q110

iran-konkur · Other · konkur-riazi_1400_specialized Sequences and series, recurrence and convergence Summation of sequence terms
110. The sequence $a_n = \begin{cases} 2^k & ; \ n = 2k \\ -2k+4 & ; \ n = 2k+1 \\ \left[\dfrac{n}{k+2}\right]+a & ; \ n = 2k+2 \end{cases}$ is defined for integer values of $n$, and is assumed. If the sum of the first 10 terms of this sequence is 19, then the value of $a_2 + a_5 + a_4 + \cdots + a_{29}$ is:
(1) $-2$ (2) zero (3) $2$ (4) $1$
\textbf{110.} The sequence $a_n = \begin{cases} 2^k & ; \ n = 2k \\ -2k+4 & ; \ n = 2k+1 \\ \left[\dfrac{n}{k+2}\right]+a & ; \ n = 2k+2 \end{cases}$ is defined for integer values of $n$, and is assumed. If the sum of the first 10 terms of this sequence is 19, then the value of $a_2 + a_5 + a_4 + \cdots + a_{29}$ is:

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(1) $-2$ \hfill (2) zero \hfill (3) $2$ \hfill (4) $1$

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