In GP n-th term is:
an = a • r ⁿ⁻¹
where
a = first term
r = common ratio
a3 = a • r ²
a3 = 54
a • r ² = 5r
a = 54 / r ²
a9 = a • r ⁸
a9 = 39366
a • r ⁸ = 39366
Replace a by 54 / r ² in this equation.
54 / r ² • r ⁸ = 39366
54 • r ⁶ = 39366
Divide both sides by 54.
r ⁶ = 729
r = six root ( 729 )
r = 3
Put this value in equation:
a • r ² = 54
a • 3 ² = 54
a • 9 = 54
a = 54 / 9
a = 6
a6 = a • r ⁵ = 6 • 3 ⁵ = 6 • 243
a6 = 1458
In GP sum of the first n terms is:
Sn = a ( 1 - r ⁿ ) / ( 1 - r )
In this case n = 10 , a = 6 , r = 3
S10= 6 • ( 1 - 3¹⁰ ) / ( 1 - 3 ) =
6 • ( 1 - 59049 ) / - 2 =
6 • ( - 59048 ) / - 2 =
- 354288 / - 2
S10 = 177144
The 3rd and the 9th terms of a G.p are 54 and 39366 respectively find (a) common ratio, (b) 1st term, (c) 6th term and (d) sum of the first 10 terms
3 answers
3rd term = ar^2 = 54
9th term = ar^8 = 39366
divide them:
r^6 = 729 = 3^6
so r = 3
if ar^2 = 54 and r = 3
9a = 54
a = 6
use your definitions to find a, (did that)
r (did that), ar^5, and sum(10)
9th term = ar^8 = 39366
divide them:
r^6 = 729 = 3^6
so r = 3
if ar^2 = 54 and r = 3
9a = 54
a = 6
use your definitions to find a, (did that)
r (did that), ar^5, and sum(10)
My typo:
It's not correct:
a • r ² = 5r
It should be written:
a • r ² = 54
It's not correct:
a • r ² = 5r
It should be written:
a • r ² = 54