Exam

Olympiad workbook

Subject

IOQM

Chapter

Functions

Olympiad workbook · IOQM · Functions

Consider the sequence of number [n+ 2 n +(1)/(2)] for n ≥ 1 , where...

Consider the sequence of number $\left[n+\sqrt{2 n}+\frac{1}{2}\right]$ for $n \geq 1$, where $[x]$ denotes the greatest integer not exceeding $x$. If the missing integers in the sequence are $n_1 < n_2 < n_3 < \ldots$ then find the $n_{12}$.

Step-by-step solution

$$\begin{aligned} & S_n=n+[\sqrt{2 n}+0.5], n \geq 1 \\ & {[\cdot]=\text { G.I.F. }}\end{aligned}$$

n	$\sqrt{2 n}$	$[\sqrt{2 n}+0.5]$	$n+[\sqrt{2 n}+0.5]$
1	1.4	1	2
2	2	2	4
3	2.4	2	5
4	2.8	3	7
5	3.1	3	8
6	3.4	3	9
7	3.7	4	11
8	4	4	12
9	4.2	4	13
10	4.4	4	14
11	4.6	5	16
12	4.8	5	17
13	5..0	5	18
14	5.2	5	19
15	5.4	5	20
16	5.6	6	22
17	5.8	6	23
18	6	6	24
19	6.1	6	25
20	6.3	6	26
21	6.4	6	27
22	6.6	7	29

By observing pattern,
$\therefore 12^{\text {th }}$ number in series $=78$