arithmetic/geometric sequence

1.
Let a and r be the first term and the common ratio of the G.P.

Sum to n terms of a G.P. = a(1 - r^n)/(1 - r)

Sum to 3 terms = 63
a(1 - r^3)/(1 - r) = 63 ...... [1]

Sum to 6 terms = 567/8
a(1 - r^6)/(1 - r) = 567/8 ...... [2]

[2]/[1] :
(1 - r^6)/(1 - r^3) = (567/8)/63
(1 + r^3)(1 - r^3)/(1 - r^3) = 9/8
1 + r^3 = 9/8
r^3 = 1/8
r = 1/2

Put r = 1/2 to [1] :
a[1 - (1/2)^3]/[1 - (1/2)] = 63
a(7/8)/(1/2) = 63
(7/8)a = (63/2)
a = 36

Sum to n terms = 2295/32
36[1 - (1/2)^n]/[1 - (1/2)] = 2295/32
1 - (1/2)^n = 255/256
(1/2)^n = 1/256
n = 8

Common ratio = 1/2
Number of terms to give a sum of 2295/32 = 8


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2.
(a)
Let a and r be the first term and the common ratio of the G.P.

Sum to n terms of a G.P. = a(r^n - 1)/(r - 1)

Sum of the first 8 terms = 257 x (Sum of the first 4 terms)
a(r^8 - 1)/(r - 1) = 257a(r^4 - 1)/(r - 1)
(r^4 + 1)(r^4 - 1) = 257(r^4 - 1)
r^4 + 1 = 257
r^4 = 256
r = ±4
Common ratio = 4 or -4

(b)
(4th term) - (2nd term) = 2

When r = 4 :
a(4)^3 - a(4) = 2
60a = 2
a = 1/30

When r = -4 :
a(-4)^3 - a(-4) = 2
-60a = 2
a = -1/30

When common ratio = 4, first term = 1/30
When common ratio = -4, first term = -1/30


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3.
(a)
0.5, 0.25, 0.125, 0.0625, ......
is a G.P with first term a = 0.5 and common ratio r = 0.5

2.5 + 2.25 + 2.125 + 2.0625 + ...... + nth term
= (2 + 0.5) + (2 + 0.25) + (2 + 0.125) + (2 + 0.0625) + ...... + nth term
= (2 + 2 + 2 + ..... + nth term) + (0.5 + 0.25 + 0.125 + ...... + nth term)
= 2n + 0.5(1 - 0.5^n)/(1 - 0.5)
= 2n + 1 - 0.5^n

(b)
101, 1 010, 10 100, 101 000, 1 010 000 , ......
is a G.P. with first term = 101 and common ratio r = 10

101 + 1 011 + 10 101 + 101 001 + 1 010 001 + ...... + nth term
= (101 + 0) + (1 010 + 1) + (10 100 + 1) + (101 000 + 1) + ...... + nth term
= (101 + 1 010 + 10 100 + ... + nth term) + (0 + 1 + 1 + 1 + 1 + ... + nthterm)
= [101(10^n - 1)/(10 - 1)] + (n - 1)
= [101(10^n - 1)/9] + (n - 1)
= (101/9)(10)^n - (101/9) - 1
= n - (110/9) + (101/9)(10)^n
= [9n - 110 + 101(10)^n]/9