MAT1581
ASSIGNMENT 1
2023
, Solution:
𝐵𝑖𝑛𝑜𝑚𝑖𝑎𝑙 𝑡ℎ𝑒𝑜𝑟𝑒𝑚:
𝑛(𝑛 − 1) 2 𝑛(𝑛 − 1)(𝑛 − 2) 3
(1 + 𝑥)𝑛 = 1 + 𝑛𝑥 + 𝑥 + 𝑥 + ⋯⋯
2! 3!
√𝑥 1
𝐿𝑒𝑡: 𝑥 = 𝑎𝑛𝑑 𝑛 =
3 3
1 1 1 1 1 1
√𝑥 3 1 √𝑥 (3) (3 − 1) √𝑥 2 (3) (3 − 1) (3 − 2) √𝑥 3
(1 + ) = 1 + ( ) ( ) + ( ) + ( ) + ⋯⋯
3 3 3 2! 3 3! 3
1
√𝑥 3 √𝑥 𝑥 5𝑥 √𝑥
(1 + ) = 1 + − + + ⋯⋯
3 9 81 2187
Solution:
2.1).
𝐹𝑜𝑟 (𝑎 + 𝑏)𝑛
𝑛
𝑇𝑘+1 = ( ) 𝑎𝑛−𝑘 𝑏𝑘
𝑘
ASSIGNMENT 1
2023
, Solution:
𝐵𝑖𝑛𝑜𝑚𝑖𝑎𝑙 𝑡ℎ𝑒𝑜𝑟𝑒𝑚:
𝑛(𝑛 − 1) 2 𝑛(𝑛 − 1)(𝑛 − 2) 3
(1 + 𝑥)𝑛 = 1 + 𝑛𝑥 + 𝑥 + 𝑥 + ⋯⋯
2! 3!
√𝑥 1
𝐿𝑒𝑡: 𝑥 = 𝑎𝑛𝑑 𝑛 =
3 3
1 1 1 1 1 1
√𝑥 3 1 √𝑥 (3) (3 − 1) √𝑥 2 (3) (3 − 1) (3 − 2) √𝑥 3
(1 + ) = 1 + ( ) ( ) + ( ) + ( ) + ⋯⋯
3 3 3 2! 3 3! 3
1
√𝑥 3 √𝑥 𝑥 5𝑥 √𝑥
(1 + ) = 1 + − + + ⋯⋯
3 9 81 2187
Solution:
2.1).
𝐹𝑜𝑟 (𝑎 + 𝑏)𝑛
𝑛
𝑇𝑘+1 = ( ) 𝑎𝑛−𝑘 𝑏𝑘
𝑘
