3. What are the terms a_(0),a_(1),a_(2) , and a_(3) of the sequence a_(n)) where a_(n) equals a) 2^n+1 ? b) (n+1)^n+1 c) lfloor n/2rfloor d) lfloor n/2rfloor +lceil n/2rceil

a) For the sequence $a_{n} = 2^{n} + 1$, we can find the terms $a_{0}, a_{1}, a_{2},$ and $a_{3}$ by substituting the values of $n$ into the formula:<br /><br />$a_{0} = 2^{0} + 1 = 1 + 1 = 2$<br /><br />$a_{1} = 2^{1} + 1 = 2 + 1 = 3$<br /><br />$a_{2} = 2^{2} + 1 = 4 + 1 = 5$<br /><br />$a_{3} = 2^{3} + 1 = 8 + 1 = 9$<br /><br />Therefore, the terms $a_{0}, a_{1}, a_{2},$ and $a_{3}$ of the sequence $\{a_{n}\}$ where $a_{n} = 2^{n} + 1$ are 2, 3, 5, and 9 respectively.<br /><br />b) For the sequence $a_{n} = (n+1)^{n+1}$, we can find the terms $a_{0}, a_{1}, a_{2},$ and $a_{3}$ by substituting the values of $n$ into the formula:<br /><br />$a_{0} = (0+1)^{0+1} = 1^{1} = 1$<br /><br />$a_{1} = (1+1)^{1+1} = 2^{2} = 4$<br /><br />$a_{2} = (2+1)^{2+1} = 3^{3} = 27$<br /><br />$a_{3} = (3+1)^{3+1} = 4^{4} = 256$<br /><br />Therefore, the terms $a_{0}, a_{1}, a_{2},$ and $a_{3}$ of the sequence $\{a_{n}\}$ where $a_{n} = (n+1)^{n+1}$ are 1, 4, 27, and 256 respectively.<br /><br />c) For the sequence $a_{n} = \lfloor \frac{n}{2} \rfloor$, we can find the terms $a_{0}, a_{1}, a_{2},$ and $a_{3}$ by substituting the values of $n$ into the formula:<br /><br />$a_{0} = \lfloor \frac{0}{2} \rfloor = \lfloor 0 \rfloor = 0$<br /><br />$a_{1} = \lfloor \frac{1}{2} \rfloor = \lfloor 0.5 \rfloor = 0$<br /><br />$a_{2} = \lfloor \frac{2}{2} \rfloor = \lfloor 1 \rfloor = 1$<br /><br />$a_{3} = \lfloor \frac{3}{2} \rfloor = \lfloor 1.5 \rfloor = 1$<br /><br />Therefore, the terms $a_{0}, a_{1}, a_{2},$ and $a_{3}$ of the sequence $\{a_{n}\}$ where $a_{n} = \lfloor \frac{n}{2} \rfloor$ are 0, 0, 1, and 1 respectively.<br /><br />d) For the sequence $a_{n} = \lfloor \frac{n}{2} \rfloor + \lceil \frac{n}{2} \rceil$, we can find the terms $a_{0}, a_{1}, a_{2},$ and $a_{3}$ by substituting the values of $n$ into the formula:<br /><br />$a_{0} = \lfloor \frac{0}{2} \rfloor + \lceil \frac{0}{2} \rceil = \lfloor 0 \rfloor + \lceil 0 \rceil = 0 + 0 = 0$<br /><br />$a_{1} = \lfloor \frac{1}{2} \rfloor + \lceil \frac{1}{2} \rceil = \lfloor 0.5 \rfloor + \lceil 0.5 \rceil = 0 + 1 = 1$<br /><br />$a_{2} = \lfloor \frac{2}{2} \rfloor + \lceil \frac{2}{2} \rceil = \lfloor 1 \rfloor + \lceil 1 \rceil = 1 + 1 = 2$<br /><br />$a_{3} = \lfloor \frac{3}{2} \rfloor + \lceil \frac{3}{2} \rceil = \lfloor 1.5 \rfloor + \lceil 1.5 \rceil = 1 + 2 = 3$<br /><br />Therefore