Find $N = 1 + frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+…}}}}}$
$begingroup$
Please show how to solve this step by step, because I don't even have an idea to begin with.
$$N = 1 + frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+...}}}}}$$
algebra-precalculus
edited Jan 13 at 17:50
KM101
6,0901525
asked Jan 13 at 16:52
LeminoLemino
183
$endgroup$
add a comment |
$begingroup$
Please show how to solve this step by step, because I don't even have an idea to begin with.
$$N = 1 + frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+...}}}}}$$
algebra-precalculus
edited Jan 13 at 17:50
KM101
6,0901525
asked Jan 13 at 16:52
LeminoLemino
183
$endgroup$
add a comment |
$begingroup$
Please show how to solve this step by step, because I don't even have an idea to begin with.
$$N = 1 + frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+...}}}}}$$
algebra-precalculus
edited Jan 13 at 17:50
KM101
6,0901525
asked Jan 13 at 16:52
LeminoLemino
183
$endgroup$
Please show how to solve this step by step, because I don't even have an idea to begin with.
$$N = 1 + frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+...}}}}}$$
algebra-precalculus
algebra-precalculus
edited Jan 13 at 17:50
KM101
6,0901525
asked Jan 13 at 16:52
LeminoLemino
183
edited Jan 13 at 17:50
KM101
6,0901525
asked Jan 13 at 16:52
LeminoLemino
183
edited Jan 13 at 17:50
KM101
6,0901525
edited Jan 13 at 17:50
KM101
6,0901525
edited Jan 13 at 17:50
KM101
6,0901525
6,0901525
asked Jan 13 at 16:52
LeminoLemino
183
asked Jan 13 at 16:52
LeminoLemino
183
asked Jan 13 at 16:52
LeminoLemino
183
183
add a comment |
add a comment |
2 Answers
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$begingroup$
You can see that the expression is found within itself. The trick is to manipulate that fact...
$$N+1 = 2 + frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+...}}}}}$$
$$N+1 = 2+frac{1}{N+1}$$
Can you solve the quadratic and continue from here?
P.S. You will get two values for $N$ by solving the quadratic. Only one of them is correct...
answered Jan 13 at 16:57
HaranHaran
1,149424
$endgroup$
$begingroup$
You can check your answer with me in the comments if you want to...
$endgroup$
– Haran
Jan 13 at 17:00
add a comment |
$begingroup$
Hint: $$N = 1 + cfrac{1}{2+cfrac{1}{2+cfrac{1}{2+cfrac{1}{2+cfrac{1}{ddots}}}}}$$
$$N -1= cfrac{1}{2+color{blue}{cfrac{1}{2+cfrac{1}{2+cfrac{1}{2+cfrac{1}{ddots}}}}}}$$
Notice that the blue part is also $N-1$.
$$N-1 = frac{1}{2+color{blue}{N-1}}$$
$$N-1 = frac{1}{N+1}$$
It should be simple enough from here. Also, note that by inspection, $N > 1$, so it should be easy to discard an extraneous solution.
answered Jan 13 at 17:02
KM101KM101
6,0901525
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add a comment |
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2 Answers
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2 Answers
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$begingroup$
You can see that the expression is found within itself. The trick is to manipulate that fact...
$$N+1 = 2 + frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+...}}}}}$$
$$N+1 = 2+frac{1}{N+1}$$
Can you solve the quadratic and continue from here?
P.S. You will get two values for $N$ by solving the quadratic. Only one of them is correct...
answered Jan 13 at 16:57
HaranHaran
1,149424
$endgroup$
$begingroup$
You can check your answer with me in the comments if you want to...
$endgroup$
– Haran
Jan 13 at 17:00
add a comment |
$begingroup$
You can see that the expression is found within itself. The trick is to manipulate that fact...
$$N+1 = 2 + frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+...}}}}}$$
$$N+1 = 2+frac{1}{N+1}$$
Can you solve the quadratic and continue from here?
P.S. You will get two values for $N$ by solving the quadratic. Only one of them is correct...
answered Jan 13 at 16:57
HaranHaran
1,149424
$endgroup$
$begingroup$
You can check your answer with me in the comments if you want to...
$endgroup$
– Haran
Jan 13 at 17:00
add a comment |
$begingroup$
You can see that the expression is found within itself. The trick is to manipulate that fact...
$$N+1 = 2 + frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+...}}}}}$$
$$N+1 = 2+frac{1}{N+1}$$
Can you solve the quadratic and continue from here?
P.S. You will get two values for $N$ by solving the quadratic. Only one of them is correct...
answered Jan 13 at 16:57
HaranHaran
1,149424
$endgroup$
You can see that the expression is found within itself. The trick is to manipulate that fact...
$$N+1 = 2 + frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+frac{1}{2+...}}}}}$$
$$N+1 = 2+frac{1}{N+1}$$
Can you solve the quadratic and continue from here?
P.S. You will get two values for $N$ by solving the quadratic. Only one of them is correct...
answered Jan 13 at 16:57
HaranHaran
1,149424
edited Jan 13 at 17:04
answered Jan 13 at 16:57
HaranHaran
1,149424
answered Jan 13 at 16:57
HaranHaran
1,149424
answered Jan 13 at 16:57
HaranHaran
1,149424
1,149424
$begingroup$
You can check your answer with me in the comments if you want to...
$endgroup$
– Haran
Jan 13 at 17:00
add a comment |
$begingroup$
You can check your answer with me in the comments if you want to...
$endgroup$
– Haran
Jan 13 at 17:00
$begingroup$
You can check your answer with me in the comments if you want to...
$endgroup$
– Haran
Jan 13 at 17:00
$begingroup$
You can check your answer with me in the comments if you want to...
$endgroup$
– Haran
Jan 13 at 17:00
add a comment |
$begingroup$
Hint: $$N = 1 + cfrac{1}{2+cfrac{1}{2+cfrac{1}{2+cfrac{1}{2+cfrac{1}{ddots}}}}}$$
$$N -1= cfrac{1}{2+color{blue}{cfrac{1}{2+cfrac{1}{2+cfrac{1}{2+cfrac{1}{ddots}}}}}}$$
Notice that the blue part is also $N-1$.
$$N-1 = frac{1}{2+color{blue}{N-1}}$$
$$N-1 = frac{1}{N+1}$$
It should be simple enough from here. Also, note that by inspection, $N > 1$, so it should be easy to discard an extraneous solution.
answered Jan 13 at 17:02
KM101KM101
6,0901525
$endgroup$
add a comment |
$begingroup$
Hint: $$N = 1 + cfrac{1}{2+cfrac{1}{2+cfrac{1}{2+cfrac{1}{2+cfrac{1}{ddots}}}}}$$
$$N -1= cfrac{1}{2+color{blue}{cfrac{1}{2+cfrac{1}{2+cfrac{1}{2+cfrac{1}{ddots}}}}}}$$
Notice that the blue part is also $N-1$.
$$N-1 = frac{1}{2+color{blue}{N-1}}$$
$$N-1 = frac{1}{N+1}$$
It should be simple enough from here. Also, note that by inspection, $N > 1$, so it should be easy to discard an extraneous solution.
answered Jan 13 at 17:02
KM101KM101
6,0901525
$endgroup$
add a comment |
$begingroup$
Hint: $$N = 1 + cfrac{1}{2+cfrac{1}{2+cfrac{1}{2+cfrac{1}{2+cfrac{1}{ddots}}}}}$$
$$N -1= cfrac{1}{2+color{blue}{cfrac{1}{2+cfrac{1}{2+cfrac{1}{2+cfrac{1}{ddots}}}}}}$$
Notice that the blue part is also $N-1$.
$$N-1 = frac{1}{2+color{blue}{N-1}}$$
$$N-1 = frac{1}{N+1}$$
It should be simple enough from here. Also, note that by inspection, $N > 1$, so it should be easy to discard an extraneous solution.
answered Jan 13 at 17:02
KM101KM101
6,0901525
$endgroup$
Hint: $$N = 1 + cfrac{1}{2+cfrac{1}{2+cfrac{1}{2+cfrac{1}{2+cfrac{1}{ddots}}}}}$$
$$N -1= cfrac{1}{2+color{blue}{cfrac{1}{2+cfrac{1}{2+cfrac{1}{2+cfrac{1}{ddots}}}}}}$$
Notice that the blue part is also $N-1$.
$$N-1 = frac{1}{2+color{blue}{N-1}}$$
$$N-1 = frac{1}{N+1}$$
It should be simple enough from here. Also, note that by inspection, $N > 1$, so it should be easy to discard an extraneous solution.
answered Jan 13 at 17:02
KM101KM101
6,0901525
edited Jan 15 at 12:47
answered Jan 13 at 17:02
KM101KM101
6,0901525
answered Jan 13 at 17:02
KM101KM101
6,0901525
answered Jan 13 at 17:02
KM101KM101
6,0901525
6,0901525
add a comment |
add a comment |
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