The second and fifth term of a geometric profession are 1 and 1/8 resecitive find the common ratio, first term and eight term
Common ratio=(1/8)/(1/4)
Common ratio=1/8*4/1
Common ratio=1/2
Sn=a(1-r^n)/1-r
S8=1/4(1-1/2)/(1-1/2)
S8=1/4(1-1/256)/1-(1/2)
S8=1/4(255/256)/1/2
S8=(255/1024)/(1/2)
S8=(255/1024)*2/1
S8=255/512.
In geometric progression:
an = a1 ∙ q ⁿ ⁻ ¹
In this case:
a2 = 1 = a1 ∙ q
a1 ∙ q = 1
a5 = 1 / 8 = a1 ∙ q⁴
a1 ∙ q⁴ = 1 / 8
Now:
a1 ∙ q⁴ = 1 / 8
a1 ∙ q ∙ q³ = 1 / 8
Since a1 ∙ q = 1
1 ∙ q³ = 1 / 8
q³ = 1 / 8
q = ∛ ( 1 / 8 )
q = ∛1 / ∛8
q = 1 / 2
Put this value in equation:
a1 ∙ q = 1
a1 ∙ 1 / 2 = 1
Multiply both sides by 2
a1 = 2
a8 = a1 ∙ q⁷
a8 = 2 ∙ ( 1 / 2 )⁷
a8 = 2 ∙ 1⁷ / 2⁷
a8 = 2 ∙ 1 / 2⁷
a8 = 1 / 2⁶
a8 = 1 / 64
Your geometric progression:
2 , 1 , 1 / 2 , 1 / 4 , 1 / 8 , 1 / 16 , 1 / 32 , 1 / 64 ....
To find the common ratio, we divide the given fifth term by the given second term.
Common Ratio (r) = 1/8 / 1 = 1/8
To find the first term (a), we use the formula:
a = second term / common ratio
a = 1 / (1/8) = 1 * (8/1) = 8
To find the eighth term, we use the formula:
nth term = a * r^(n-1)
where n is the term number we want to find.
eighth term = a * r^(8-1) = 8 * (1/8)^(7) = 8 * (1/8)^7 = 8 * (1/8)*(1/8)*(1/8)*(1/8)*(1/8)*(1/8)*(1/8) = 1/128
Therefore, the common ratio is 1/8, the first term is 8, and the eighth term is 1/128.
a_5/a_2 = ar^4/ar = r^3 = 1/8
so r = 1/2
now you want a and ar^7
or, knowing r,
a = a_2/r
a_8 = a_5 * r^3