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2.2.E: Problemas sobre los números naturales y la inducción (ejercicios)

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    114136
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    Ejercicio\(\PageIndex{1}\)

    Complete los detalles faltantes en Ejemplos\((\mathrm{a}),(\mathrm{b}),\) y\((\mathrm{d})\).

    Ejercicio\(\PageIndex{2}\)

    Demostrar Teorema 2 en detalle.

    Ejercicio\(\PageIndex{3}\)

    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}
    \]

    Ejercicio\(\PageIndex{4}\)

    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.

    Ejercicio\(\PageIndex{5}\)

    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\(\varepsilon<1 ; n=1,2,3, \ldots\)

    Ejercicio\(\PageIndex{6}\)

    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\(n ]\).

    Ejercicio\(\PageIndex{7}\)

    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).

    Ejercicio\(\PageIndex{8}\)

    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\(b ?\)

    Ejercicio\(\PageIndex{9}\)

    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.

    Ejercicio\(\PageIndex{10}\)

    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)\(\sum_{k=1}^{n} k^{4}=\frac{1}{30} n(n+1)(2 n+1)\left(3 n^{2}+3 n-1\right)\).


    2.2.E: Problemas sobre los números naturales y la inducción (ejercicios) is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.