Question

Va rooog urgeent!!!!???? Dau coroana!!
6.
a) Aratati ca 1/n-1/n+1=1/n*(n+1), pentru orice n=N*.
b)Calculati:1/1*2+1/2*3+1/3*4+...+1/2013*2014.
c)Aratati ca 1/n-1/n+m=m/n*(n+m), pentru orice n,m=N*
d)Aratati ca :3/2*5+3/5*8+3/8*11+....+3/2012*2013=2013/4030.
e)Calculati: 4/1*5+4/5*9+4/9*13+...+4/2013*2017.

Answer 1

6. a. 1/n -1/n+1=n+1-n/n(n+1)=1/n(n+1)=1/n(n+1) adevarat pentru orice n∈N.
b. 1/1·2 +1/2·3 +1/3·4 +......+ 1/2013·2014=
1/1-1/2+1/2-1/3+1/3-1/4+.......+1/2013-1/2014=
1/1-1/2014=2013/2014;
c. 1/n -1/n+m=n+m-n/n(n+m)=m/n(n+m)=m/n(n+m) adevarat pentru orice n;m∈N.
d. 3/2·5 +3/5·8 +3/8·11 +...+3/2012·2013=
3/2-3/5+3/5-3/8+3/8-3/11+3/11-.......+3/2012-3/2013=
3/2-3/2013=3·2013-2·3/4026=3·2011/4026=4033/4026=4033/4026≠2013/4030(cred ca n-ai transcris bine enuntul);
e. 4/1·5 +4/5·9 +4/9·13+......+4/2013·2017=
4/1-4/5+4/5-4/9+4/9-4/13+4/13-........+4/2013-4/2017=
4/1-4/2017=8064/2017;