2.2.E: Problemas sobre los números naturales y la inducción (ejercicios)
- Page ID
- 114136
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Demostrar Teorema 2 en detalle.
Supongamos\(x_{k}<y_{k}, k=1,2, \ldots,\) en un campo ordenado. Demostrar por inducción en\(n\) ese
\(\sum_{k=1}^{n} x_{k}<\sum_{k=1}^{n} y_{k}\)
(a) (b) si todos\(x_{k}, y_{k}\) son mayores que cero, entonces
\ [\ prod_ {k=1} ^ {n} x_ {k} <\ prod_ {k=1} ^ {n} y_ {k}
\]
Demostrar por inducción que
(i)\(1^{n}=1\);
(ii)\(a<b \Rightarrow a^{n}<b^{n}\) si\(a>0\).
De ahí deducir que
(iii)\(0 \leq a^{n}<1\) si\(0 \leq a<1\);
(iv)\(a^{n}<b^{n} \Rightarrow a<b\) si\(b>0 ;\) prueba por contradicción.
Demostrar las desigualdades de Bernoulli: Para cualquier elemento\(\varepsilon\) de un campo ordenado,
(i)\((1+\varepsilon)^{n} \geq 1+n \varepsilon\) si\(\varepsilon>-1\);
(ii)\((1-\varepsilon)^{n} \geq 1-n \varepsilon\) si\(\varepsilon<1 ; n=1,2,3, \ldots\)
Para cualquier elemento de campo\(a, b\) y números naturales\(m, n,\) prueben que
\ [\ begin {array} {ll} {\ text {(i)} a^ {m} a^ {n} =a^ {m+n};} & {\ text {(ii)}\ left (a^ {m}\ right) ^ {n} =a^ {m n}}\\ {\ text {(iii)} (a b) ^ {n} =a^ {n} b^ {n};} & {\ text {(iv)} (m+n) a=m a+n a}\\ {\ text {( v)} n (m a) = (n m)\ cdot a;} & {\ text {(vi)} n (a+b) =n a+n b}\ end {array}
\]
[Pista: Para problemas que involucran dos números naturales, arregla\(m\) y usa la inducción en\(n ]\).
Demostrar que en cualquier campo,
\ [
a^ {n+1} -b^ {n+1} =( a-b)\ sum_ {k=0} ^ {n} a^ {k} b^ {n-k},\ quad n=1,2,3,\ ldots
\]
De ahí para\(r \neq 1\)
\ [\ sum_ {k=0} ^ {n} a r^ {k} =a\ nfrac {1-r^ {+1}} {1-r}
\]
(suma de \(n\)términos de una serie geométrica).
Para\(n>0\) define
\ [
\ left (\ begin {array} {l} {n}\\ {k}\ end {array}\ right) =\ left\ {\ begin {array} {ll} {\ frac {n!} {k! (n-k)!} ,} & {0\ leq k\ leq n}\\ {0,} & {\ text {de lo contrario}}\ end {array}\ right.
\]
Verifica la ley de Pascal,
\ [\ left (\ begin {array} {l} {n+1}\\ {k+1}\ end {array}\ right) =\ left (\ begin {array} {l} {n}\\ {k}\ end {array}\ right) +\ left (\ begin {array} {c} {n}\\ {k+1}\ end {array}\ derecha).
\]
Luego pruebe por inducción en\(n\) eso
(i)\((\forall k | 0 \leq k \leq n)\left(\begin{array}{l}{n} \\ {k}\end{array}\right) \in N ;\) y
(ii) para cualquier elemento de campo\(a\) y\(b\),
\ [
(a+b) ^ {n} =\ sum_ {k=0} ^ {n}\ left (\ begin {array} {l} {n}\\ {k}\ end {array}\ right) a^ {k} b^ {n-k} ,\ quad n\ in N\ text {(el teorema binomial).}
\]
Qué valor debe\(0^{0}\) tomar para que (ii) se mantenga para todos\(a\) y\(b ?\)
Mostrar por inducción que en un campo ordenado\(F\) cualquier secuencia finita\(x_{1}, \ldots, x_{n}\) tiene un término más grande y uno menor (que no necesita ser\(x_{1}\) o\(x_{n} ) .\) Deducir que todo\(N\) es un conjunto infinito, en cualquier campo ordenado.
Demostrar en\(E^{1}\) ello
(i)\(\sum_{k=1}^{n} k=\frac{1}{2} n(n+1)\);
(ii)\(\sum_{k=1}^{n} k^{2}=\frac{1}{6} n(n+1)(2 n+1)\);
(iii)\(\sum_{k=1}^{n} k^{3}=\frac{1}{4} n^{2}(n+1)^{2}\);
(iv)\(\sum_{k=1}^{n} k^{4}=\frac{1}{30} n(n+1)(2 n+1)\left(3 n^{2}+3 n-1\right)\).