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Albert1
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A sequence {$a_n$},$S_n=a_1+a_2+-----+a_n$,for each $n\in N, a_n>0$
if $2S_n=a_n+\dfrac {1}{a_n}$
please find $a_n=?$ (express in $n$)
if $2S_n=a_n+\dfrac {1}{a_n}$
please find $a_n=?$ (express in $n$)
Albert said:A sequence {$a_n$},$S_n=a_1+a_2+-----+a_n$,for each $n\in N, a_n>0$
if $2S_n=a_n+\dfrac {1}{a_n}$
please find $a_n=?$ (express in $n$)
thanks greg1313:greg1313 said:I claim the sequence is $1,\sqrt2-1,\sqrt3-\sqrt2,\sqrt4-\sqrt3,\dots$ and thus the general term is
$$a_n=\sqrt n-\sqrt{n-1}$$
$$2S_n=a_n+\dfrac{1}{a_n}\implies S_n=\dfrac{a^2_n+1}{2a_n}$$
Proof by induction:
We have
$$1+(\sqrt2-1)+(\sqrt3-\sqrt2)+\dots+\sqrt{n}-\sqrt{n-1}=\sqrt n$$
$$S_{n+1}=\dfrac{(\sqrt{n+1}-\sqrt{n})^2+1}{2(\sqrt{n+1}-\sqrt{n})}$$
$$=\dfrac{n+1-2\sqrt{n+1}\sqrt{n}+n+1}{2(\sqrt{n+1}-\sqrt{n})}$$
$$=\dfrac{2(n+1)-2\sqrt{n+1}\sqrt{n}}{2(\sqrt{n+1}\sqrt{n})}$$
$$=\dfrac{2\sqrt{n+1}(\sqrt{n+1}-\sqrt{n})}{2(\sqrt{n+1}-\sqrt{n})}=\sqrt{n+1}$$
It follows that $a_n=\sqrt{n}-\sqrt{n-1}$.
Albert said:thanks greg1313:
very good!your answer is correct
$given: \,\, 2S_n=a_n+\dfrac{1}{a_n}---(1)\\$
$for \,\,n=1\\$
$S_1=a_1=1\\$
$for \,\,n\geq 2\\$
$a_n=S_n-S_{n-1}---(2)\\$
$put \,\, (2)\,\, to \,\,(1)\\$
we have :$S_n^2-S_{n-1}^2=1\,\,\therefore S_n^2=1+n-1=n\\$
$\therefore S_n=\sqrt n$
from (2):$a_n=\sqrt{n}-\sqrt{n-1}$ #
A sequence is a list of numbers that follow a specific pattern or rule.
$a_n$ stands for the nth term in a sequence, where n represents the position of the term in the sequence.
To find $a_n$ in a sequence, you need to first identify the pattern or rule that the sequence follows. Then, you can use this pattern to find the missing term at position n in the sequence.
Yes, $a_n$ can be a decimal or fraction depending on the pattern or rule of the sequence. Some sequences may have all whole numbers, while others may have decimals or fractions.
Yes, there are different methods for finding $a_n$ in a sequence depending on the type of sequence. Some common methods include using a formula, creating a table, or finding a recursive relationship.