Answer to Question #223135 in Real Analysis for Darren Agbor

Question #223135

The sum of the first n terms of a sequence (Un) n € N* is given by Sn = n(n+1) / n+2. Find

a) The nth term of the sequence

b) The sum of the terms from the 6th to the 31st term inclusive and exclusive.



1
Expert's answer
2021-09-23T00:33:47-0400

a)


Sn=Sn1+an,n2S_n=S_{n-1}+a_n, n\geq2

an=SnSn1=n(n+1)n+2(n1)(n1+1)n1+2a_n=S_n-S_{n-1}=\dfrac{n(n+1)}{n+2}-\dfrac{(n-1)(n-1+1)}{n-1+2}

=n(n2+2n+1n22n+n+2)(n+1)(n+2)=\dfrac{n(n^2+2n+1-n^2-2n+n+2)}{(n+1)(n+2)}

=n(n+3)(n+1)(n+2),n2=\dfrac{n(n+3)}{(n+1)(n+2)}, n\geq2

a1=S1=1(1+1)1+2=23a_1=S_1=\dfrac{1(1+1)}{1+2}=\dfrac{2}{3}

b)


S5=5(5+1)5+2=307S_5=\dfrac{5(5+1)}{5+2}=\dfrac{30}{7}

S6=6(6+1)6+2=214S_6=\dfrac{6(6+1)}{6+2}=\dfrac{21}{4}

S30=30(30+1)30+2=46516S_{30}=\dfrac{30(30+1)}{30+2}=\dfrac{465}{16}

S31=31(31+1)31+2=99233S_{31}=\dfrac{31(31+1)}{31+2}=\dfrac{992}{33}

The sum of the terms from the 6th to the 31st term inclusive


S31S5=99233307=5954231S_{31}-S_5=\dfrac{992}{33}-\dfrac{30}{7}=\dfrac{5954}{231}

The sum of the terms from the 6th to the 31st term exclusive


S30S6=46516214=38116S_{30}-S_6=\dfrac{465}{16}-\dfrac{21}{4}=\dfrac{381}{16}

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