Question

Determinați numărul natural n pentru care : 
a) $8^{n}+ 8^{n+1} =18* 2^{2003}$
b)$9^{n}+ 9^{n+1}=10* 3^{2012}$
c)[tex] 6^{n}+ 6 ^{n+3}=217* 6^{55}
[/tex]
d)$7^{n+1} + 7^{n+2} =8*7^{11}$

Answer 1

a)
$8^{n}$(1+8)=18*2²°°³

$8^{n}$*9=18*2²°°³    I:9
2³^n=2*2²°°³
$2^{3+n}$=2²°°⁴
3+n=2004
n=2001


b)
$9^{n}$(1+9)=10*3²°¹²


$9^{n}$*10=10*3²°¹²  I:10


$9^{n}$=3²°¹²
3²^n=3²°¹²
$3^{2+n}$=2012

2+n=2012
n=2010



c)
$6^{n}$*(1+6³)=217*6⁵⁵

$6^{n}$*217=217*6⁵⁵  I:217

$6^{n}$=6⁵⁵

n=55



d)

$7^{n+1}$*(1+7)=8*7¹¹


$7^{n+1}$*8=8*7¹¹  I:8


$7^{n+1}$=7¹¹

n+1=11
n=10