Question

Calcule a soma dos termos da PG finita. a) 1,2,..., 512); b) (5,20,..., 1280);c) (1,2^2,...,2^10).

Answer 1

a1 = 1
a2 = 2
q = 2/1 = 2 ***
an = 512
512 = 1 * 2ⁿ⁻¹
2ⁿ⁻¹ = 512  =  2⁹
n - 1 = 9
n = 10 ****
S10 = 1 ( 2¹⁰  - 1) ( 2 - 1)
S10 = ( 1024 - 1)
s10 = 1023

2
a1 = 5
a2 = 20
an = 1280
q = 20/5 = 4 ****
an = 5 * 4ⁿ⁻¹
1280 = 5 * 4ⁿ⁻¹
4ⁿ⁻¹  = 1280/5 = 256 =  2⁸
(2²)ⁿ⁻¹ = 2⁸
(2)²ⁿ⁻² = 2⁸
2n - 2 = 8
2n = 8 + 2
2n = 10
n = 5 *****
S5 =5 ( 4⁵  -  1)/( 4 - 1)
S5 = 5 ( 1024 - 1)/3
s5 = 5 ( 1023)/3
S5 = 5115/3 =1705

3
a1 = 1
a2 = 2²
an = 2¹⁰
q = 2² /1 = 2² ou 4
2¹⁰ = 1 *( 2²)ⁿ⁻¹
2¹⁰ = (2 )²ⁿ⁻²
2n - 2 = 10
2n = 10 + 2
2n = 12
n = 6 *****

S6 = [1 ( 4)⁶  - 1 ]/ ( 4 - 1)
S6 = [ 4096 - 1]/3
S6 = 4095 /3 
S6 = 1365 ***