Respuesta :
Good morning ☕️
Answer:
The expression 1!+2!+3!+4!+.....+n! (Where n is a natural number) is not a / an:
c) perfect square
Step-by-step explanation:
_______1!+2!+3!+4!+.....+n! Is a composite number____________
proof: 1!+2!=3 and 3!+4!+.....+n! has 3 as a factor
then 1!+2!+3!+4!+.....+n! has 3 as a factor
__________________________________________________
_______1!+2!+3!+4!+.....+n! is an odd number________________
proof: 2!+3!+4!+.....+n! has 2 as a factor then it’s an even number
then if we add 1! , we get an odd number 1!+2!+3!+4!+.....+n!
__________________________________________________
______1!+2!+3!+4!+.....+n! is a multiple of 3__________________
proof : 1!+2!=3 and 3!+4!+.....+n! has 3 as a factor
then 1!+2!+3!+4!+.....+n! has 3 as a factor therefore it’s a multiple of 3.
__________________________________________________
______1!+2!+3!+4!+.....+n! is not a perfect square _____________
proof: a perfect square number must end with one of these digits 0,1,4,5,6 and 9
in other words it will never end with 3
reason :
0^2=0
1^2=1
2^2=4
3^2=9
4^2=16
5^2=25
6^2=36
7^2=49
8^2=64
9^2=81
in addition, the factorial of any number from 5 to n end with zero
cause it has 2 and 5 as factors at the same time
examples:
5!=120
6!=720
7!=5 040
.
.
.
then the sum 5!+.....+n! Will end with 0
also, 1!+2!+3!+4!=33
conclusion:
1!+2!+3!+4!+.....+n! = (1!+2!+3!+4!) +(5! +.....+n!)
= 33 + ( a large number which end with 0)
= a large number that ends with 3
we conclude that :1!+2!+3!+4!+.....+n! is not a perfect square
__________________________________________________
:)
Answer:
c) perfect square
Step-by-step explanation:
Lets take one example:
n = 4
1 + 2! + 3! + 4! = 33
33 is odd, multiple of 3 and also composite. But, not a perfect square
