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

Cezar

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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:

. . . . .\(\displaystyle a_1\, \sqrt{\strut a_1\,}\, +\, a_2\, \sqrt{\strut a_2\, }\, +\, ...\, +\, a_n\, \sqrt{\strut a_n\,}\, =\)

Last step from the answer:

. . . . .\(\displaystyle a_1\, -\, a_2\, +\, a_2\, -\, a_3\, +\, ...\, +\, a_n\, -\, a_{n+1}\, =\)

. . . . .\(\displaystyle a_1\, -\, a_n\)

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

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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:

. . . . .\(\displaystyle a_1\, \sqrt{\strut a_1\,}\, +\, a_2\, \sqrt{\strut a_2\, }\, +\, ...\, +\, a_n\, \sqrt{\strut a_n\,}\, =\)

Last step from the answer:

. . . . .\(\displaystyle a_1\, -\, a_2\, +\, a_2\, -\, a_3\, +\, ...\, +\, a_n\, -\, a_{n+1}\, =\)

. . . . .\(\displaystyle a_1\, -\, a_n\)

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

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

And, solving with the second formula, knowing that

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

...we get:

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

Further:

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

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

And that would be the formula.



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

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