Question-192367

Question Number 192367 by Rupesh123 last updated on 15/May/23

Answered by mehdee42 last updated on 15/May/23

t_{r} = r - \frac{3}{2} \Rightarrow t_{r + 1} = r - \frac{1}{2} & t_{r + 2} = r + \frac{1}{2} & t_{r + 3} = r + \frac{3}{2}

\Rightarrow t_{r} t_{r + 1} t_{r + 2} t_{r + 3} = r^{4} - \frac{5}{2} r^{2} + \frac{25}{26}

\Rightarrow \underset{r = 1}{\sum^{n}} \sqrt{1 + t_{r} t_{r + 1} t_{r + 2} t_{r + 3}} = \underset{r = 1}{\sum^{n}} \sqrt{r^{4} - \frac{5}{2} r^{2} + \frac{25}{16}}

= \underset{r = 1}{\sum^{n}} (r^{2} - \frac{5}{4}) = \frac{n (n + 1) (2 n + 1)}{6} - \frac{5}{4} n ✓

Commented by Rupesh123 last updated on 15/May/23

Nice, sir!