Question

Calculati :
a) 2^1•2^2+2^3+...+2^100
b)3•3^2•3^3•...•3^201
c)4^1•4^3•4^5•...•4^51
d)2+2^2+2^3+...+2^100
e)3^2+3^3+3^4+...+3^108
f)10^10+10^11+10^12+...+102014
*DAU 55 PUNCTE*

Answer 1

a)
2^1×2^2×2^3×...×2^100 =
= 2^(1+2+3+...+100) =
= 2^(100×101:2) =
= 2^(50×101) =
= 2^5050

b) 3^1 × 3^2 × 3^3 ×...×3^201 =
= 3^(1+2+3+...+201) =
= 3^(201×202: 2) =
= 3^(201×101) =
= 3^20301

c) 4^1 × 4^3 × 4^5 ×...×4^51 =
= 4^(1+3+5+...+51) =
= 4^(26×52:2) =
= 4^676

d) S = 2 + 2^2 + 2^3 +...+ 2^100 |×2
2×S = 2^2 + 2^3 +2^4 +...+2^101
2×S - S = 2^101 - 2
S = 2^101 - 2

e) S = 3^2 + 3^3 + 3^4 +...+ 3^108 |×3
3×S = 3^3 + 3^4 + 3^5 +...+ 3^109
3×S - S = 3^109 - 3^2
2 ×S = 3^109 - 9
S = (3^109 -9) : 2

f) S = 10^10 + 10^11 + 10^12 +...+ 10^2014 |×10
10×S = 10^11 + 10^12 + 10^13 +...+10^2015
10×S - S = 10^2015 - 10^10
9×S = 10^10 (10^2005 - 1)
S = 10^10(10^2005 - 1) : 9