Question

Determinati ultima cifra a numerelor:
a)b=7+7^2+7^3+...+7^43
b)c=3^0+3^1+3^2+3^3+...3^42
c)d=8+8^1+8^2+8^3+...+8^88
Ps: ^=la puterea

Answer 1

a) S= (8 + 8⁴³) *43/2

8¹=(.....8)
8²=(.....4)
8³=(......2)
8⁴=(.....6)
8⁵=(.......8)

se repeta din 4 in 4

43:4=10 rest 3  =>8⁴³=(.....2)

S=[8+(......2)] *43/2=  (......10)*43/2=  (.....5)*43=  (.......5)

b)S=3⁰ +3+3² +....+3⁴²
S=1+3+3²+...+3⁴²

S=(1+3⁴²)*43/2

3¹=(.....3)
3²=(.....9)
3³=(.....7)
3⁴=(.....1)
3⁵=(.....3)

se repeta din 4 in 4

42:4=10 rest 2  =>3⁴²=(.....9)

S=[1+(....9)]*43/2=  (.....10)*43/2=  (.....5)*43=  (.....5)

c)S=8+8¹+8²+....+8⁸⁸

S=(8+8⁸⁸)*89/2

8¹=(.....8)
8²=(.....4)
8³=(.....2)
8⁴=(.....6)
8⁵=(.....8)

se repeta din 4 in 4  88:44=2  =>8⁸⁸=(......8)

S=[8+(.....8)]*89/2=  (.......16)*89/2=  (....8)*89=  (.....2)