Question

Ex. 17 va rog rapid dau multe puncte

Answer 1

a)  $2^{1+2+3+...+12} =2^{n+3}$
=> 1+2+3+...+12=n+3
12*13:2=n+3
78=n+3
n=75

b) $7^{11+12+13+...+30}=(7^n)^3*49$
11+12+13+...+30=(30+11)(30-11+1):2=41*20:2=410
$7^{410}=7^{3n}*7^2$
$7^{408}*7^2=7^{3n}*7^2$
$7^{408}=7^{3n}$
3n=408
n=136

c)$3^{2+4+6+...+20}=3^{n(n+1)}$
2+4+6+...+20=2(1+2+3+...+10)=2*10*11:2=10*11
$3^{10*11}=3^{n(n+1)}$
10*11=n(n+1)
=> n=10

d)$5^{20+24+28+...48}=25^n$
20+24+28+...+48=4(5+6+7+...+12)=4*(12+5)(12-5+1):2=4*17*8:2=2*136
$5^{2*136}=5^{2n}$
2*136=2n
n=136

e)$11^{1+3+5+...+31}=11^{n^2}$
1+3+5+...+31=(31+1)[(31-1):2+1]:2=32*16:2=256
$11^{256}=11^{n^2}$
n²=256
n=16

f)$2^{5+12+19+...+75}=2^{11n}$
5+12+19+...+75=(75+5)[(75-5):7+1]:2=80*11:2=40*11
$2^{40*11}=2^{11n}$
40*11=11n
n=40