92. (a) Show that if lim,→» A2n = L and lim,- n A2n+1 = L, = L, then {a,} is convergent and lim, (b) If a = 1 and no An = L. an+1 = 1 + 1 + an find the first eight terms of the sequence {a,}. Then use part (a) to show that lim,o An = /2. This gives the continued fraction expansion V2 = 1 + 1 2 + 2 + .
Percentage
A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.
Algebraic Expressions
In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.
Numbers
Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.
Subtraction
Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.
Addition
Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.
![onacci
Repeat this process so that, in general,
-2
a, + b,
Ja,b,
an+1 =
bn+1 =
(a) Use mathematical induction to show that
a, > an+1 > b,+1 > bn
and
(b) Deduce that both {a,} and {b»} are convergent.
(c) Show that lim,,- an
lim, bn. Gauss called the
||
common value of these limits the arithmetic-geometric
mean of the numbers a and b.
92. (a) Show that if lim, A2n = L and lim, a2n+1 = L,
then {a,} is convergent and lim,→« An = L.
(b) If a1
n→∞
= 1 and
nd
1
an+1 = 1 +
1+ an
find the first eight terms of the sequence {a,}. Then use
part (a) to show that lim,o An =
continued fraction expansion
2. This gives the
n.]
1
V2 = 1 +
2 +
2 + ..
123c0
93. The size of an undisturbed fish population has been modeled by
the formula
bp
Pn+1 =
a + Pn
where p, is the fish population after n years and a and b are
positive constants that denend on the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcba78722-b839-46bb-b46d-8bfd06aff38d%2F059d4023-3196-4f3d-82f3-be47657b2156%2Fe7i5tos_processed.jpeg&w=3840&q=75)
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