Question

a)3^10-2×3^9-2×3^8-2×3^7-...2×3^1-2×3^0; b) 2^10-2^9-2^8-2^7-...-2-1; c) 5^100-4×5^99-4×5^98-....-4×5^2-4×5-4×5^0; d) 7^20-6×7^19-6×7^18-...6×7^2-6×7.

Answer 1

a)3¹⁰-2×3⁹ -2×3⁸-2×3⁷-...2×3¹-2×3⁰= 3¹⁰-2*(3⁹+3⁸+3⁷+3⁶+3⁵+3⁴+3²+3¹+3⁰)=
= 3¹⁰- 2*(3¹⁰- 1)(3-1)=3¹⁰- 2*(3¹⁰- 1)/2=3¹⁰-3¹⁰+1= 1
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b)2¹⁰-2⁹-2⁸-2⁷-2⁶-....-2-1= 2¹⁰- (2⁹+2⁷+2⁶+.....+2+1)=2¹⁰-(2¹⁰ - 1)=2¹⁰ -2¹⁰+1=1

c)5¹⁰⁰ -4*5⁹⁹ -4*5⁹⁸-4*5⁹⁷-......-4*5-4*5⁰=5¹⁰⁰- (4*5⁹⁹ +4*5⁹⁸+4*5⁹⁷+....+4*5+4*5⁰)=
=5¹⁰⁰-4*(5⁹⁹+5⁹⁸+5⁹⁷+.....+5¹+5⁰)=5¹⁰⁰ -4*(5¹⁰⁰ - 1) /4= 5¹⁰⁰ - 5¹⁰⁰ +1= 1

d) 7²⁰- 6*7¹⁹- 6*7¹⁸-6*7¹⁷-....-6*7¹ -6*7⁰=7²⁰-6* (7¹⁹+7¹⁸+7¹⁷+...+7¹+7⁰)=
=7²⁰- 6*(7²⁰- 1)/6= 7²⁰- (7²⁰-1)= 7²⁰-7²⁰+1= 1