Thread: Sum of n terms of sequence: a_1 sqrt[a_1] + a_2 sqrt[a_2] + ... + a_n sqrt[a_n]

1. Sum of n terms of sequence: a_1 sqrt[a_1] + a_2 sqrt[a_2] + ... + a_n sqrt[a_n]

Hi, I have an exercise that would be considered to belong to calculus, but i got stuck on solving a sum(even further, i have the final step at the answer, but I don't understand how they got to the conclusion).
Basically i have this sum, but i don't know the steps between where i got stuck and the last step from the answer.

Where I got stuck:

. . . . .$a_1\, \sqrt{\strut a_1\,}\, +\, a_2\, \sqrt{\strut a_2\, }\, +\, ...\, +\, a_n\, \sqrt{\strut a_n\,}\, =$

. . . . .$a_1\, -\, a_2\, +\, a_2\, -\, a_3\, +\, ...\, +\, a_n\, -\, a_{n+1}\, =$

. . . . .$a_1\, -\, a_n$

I wold really appreciate if someone could explain me the steps in between.

2. Originally Posted by Cezar
Hi, I have an exercise that would be considered to belong to calculus, but i got stuck on solving a sum(even further, i have the final step at the answer, but I don't understand how they got to the conclusion).
Basically i have this sum, but i don't know the steps between where i got stuck and the last step from the answer.

Where I got stuck:

. . . . .$a_1\, \sqrt{\strut a_1\,}\, +\, a_2\, \sqrt{\strut a_2\, }\, +\, ...\, +\, a_n\, \sqrt{\strut a_n\,}\, =$

. . . . .$a_1\, -\, a_2\, +\, a_2\, -\, a_3\, +\, ...\, +\, a_n\, -\, a_{n+1}\, =$

. . . . .$a_1\, -\, a_n$

I wold really appreciate if someone could explain me the steps in between.
Did they provide any relationship between a1 and a2 and a3 ..... an?

3. Originally Posted by Subhotosh Khan
Did they provide any relationship between a1 and a2 and a3 ..... an?
Thank you for response. Apparently I've done the exercise with another formula that I was supposed to do. Basically when I was supposed to study the monotony of the sequence I used formula 1) instead of 2), and I haven't seen the relation from the second formula.

1) If $\dfrac{a_{n+1}}{a_n}\, \geq\, 1,$ then the sequence is increasing; otherwise, it is decreasing.

2) If $a_{n+1}\, -\, a_n\, \geq\, 0,$ then the sequence is increasing; otherwise, it is decreasing.

And, solving with the second formula, knowing that

. . .$a_{n+1}\, =\, a_n\, \left(1\, -\, \sqrt{\strut a_n\,}\right)$

...we get:

. . .$a_{n+1}\, -\, a_n\, =\, a_n\, \left(1\, -\, a_n\right)\, -\, a_n\, =\, -a_n\, \sqrt{\strut a_n\,}$

Further:

. . .$a_{n+1}\, -\, a_n\, =\,-a_n\, \sqrt{\strut a_n\,}\, \big|\, \cdot\, -1$

. . .$a_n\, \sqrt{\strut a_n\,}\, =\, a_n\, -\, a_{n+1}$

And that would be the formula.

Thank you for your time, I guess next time I'll try all formulas before asking