1+ 3+ 5 + ... + 37=
1 + (2+ 1)+ ( 4+ 1)+ ... + ( 36+1)=
(1+ 1+ ... + 1)+ (2+ 4+ ... +36) =
(1+ 1+ ... + 1)+ (2·1+ 2·2+ ... +2·18) =
(1+ 1+ ... + 1)+ [2·( 1+ 2+ ... + 18)] =
1·19 + 2·18·(18+ 1):2=
19+ 18·19=
19·1+ 18·19=
19·( 1+ 18) =
19·19=
361
1+ 3+ 3² + 3³ + ... + 3²⁰ = ?
S= 1+ 3+ 3² + 3³ + ... + 3²⁰ I ·3
3S= 3+ 3²+ 3³+ 3⁴+ ... + 3²¹
____________________________
3S - S = 3+ 3²+ 3³+ 3⁴+ ... + 3²¹ -
1- 3- 3² - 3³ - ... - 3²⁰
___________________________
2S = 3²¹ - 1
S = _3²¹ - 1_
2
