Răspuns:
Explicație pas cu pas:
a) 2¹⁺²⁺³⁺........⁺¹⁶ = 4ⁿ⁺⁷
Aducem egalitatea la aceeasi baza:
<=> 2¹⁺²⁺³⁺........⁺¹⁶ = (2²)ⁿ⁺⁷ <=>
2¹⁺²⁺³⁺............¹⁶ = 2²⁽ⁿ⁺⁷⁾ =>
1+2+3+.......+16 = 2n+14 , Folosim formula lui Gauss =>
(1+16)·16:2 = 2n + 14 <=> 17·8 -14 = 2n =>
2n = 136-14 <=> 2n = 122 => n = 122:2 => n = 61
b) 2⁵⁺¹²⁺¹⁹............⁺⁷⁵ = 16¹¹ⁿ <=>
2⁵⁺¹²⁺¹⁹............⁺⁷⁵ = (2⁴)¹¹ⁿ <=>
2⁵⁺¹²⁺¹⁹............⁺⁷⁵ = 2⁴⁴ⁿ =>
5+12+19..........+75 = 44n =>
(5+75)·[(75-5):7+1]:2 = 44n =>
80·11:2 = 44n <=> 40·11 = 44n I : 11 =>
40 = 4n => n = 40:4 => n = 10
c) 15ⁿ⁺⁸ : 5ⁿ⁺⁸ = 81⁶ <=>
(15:5)ⁿ⁺⁸ = (3⁴)⁶ <=> 3ⁿ⁺⁸ = 3⁴ˣ⁶ =>
n+8 = 24 => n = 24-8 => n = 16
d) 4²⁺⁴⁺⁶⁺..............⁺²⁰ = 4ⁿ⁽ⁿ⁺¹⁾ =>
2+4+6+..........+20 = n(n+1) <=>
2(1+2+3+........+10) = n(n+1) =>
2·11·10:2 = n(n+1) <=>
110 = n(n+1) , produs de doua numere consecutive =>
n = 10 ; n+1 = 11 => n = 10
