Question

quanto vale a soma dos termos da PG finita: a) ( 1, 2, ..., 2048) b) ( 5 , 20 , ..., 20480)

Answer 1

a1 = 1
a2 = 2
an = 2048
q = 2/1 = 2 ***
2048 = 1 * 2ⁿ⁻¹
2ⁿ⁻¹  = 2048  = 2¹¹
n - 1 = 11
n = 11 + 1 = 12 ****
S12 = 1 ( 2¹²  - 1)/( 2 - 1)
s12 = ( 4096 - 1)/1
S12 = 4095 ****

b
a1 = 5
a2 =20
an =20480
q = 20/5 = 4 ****
20480 = 5 * 4ⁿ⁻¹
4ⁿ⁻¹  = 20480/5 = 4096
(2²)ⁿ⁻¹ = 4096 = 2¹²
( 2)²ⁿ⁻² = 2¹²
2n - 2 = 12
2n = 12 + 2 
2n = 14
n = 7 ****
S7 = 5 ( 4⁷  - 1 )/ ( 4 - 1)
S7 = 5 ( 256 - 1 ) / 3
S7 = 5 ( 255)/ 3
S7 = 1275/3 =425 ****