Question 1201412
the formula for the geometric series is An = A1 * r ^ (n-1)
An is the nth term.
A1 is the first term
r ^ (n-1) is the common multiplier raised to the (n-1)th term.
in your problem, the seventh term would be A7 and the ninth term would be 9
you will have:
A7 = 1458
A9 = 13122
the formula for the 7th term is A7 = A1 * r ^ 6
the formula for the 9th term is A9 = A1 * r ^ 8
replace A7 and A9 with their respective values to get:
1458 = A1 * r ^ 6
13122 = A1 * r ^ 8
solve for A1 in both equations to get:
A1 = 1458 / r^6
A1 = 13122 / r^8
since they are both equal to A1, you get:
1458 / r^6 = 13122 / r^8
multiply both sides of the equation by r^8 and divide both sides of the eqution by 1458 to get:
r^8 / r^6 = 13122 / 1458
simplify to get:
r^2 = 9
solve for r to get:
r = 3
replace r in the first equation to get:
1458 = A1 * 3 ^ 6
solve for A1 to get:
A1 = 1458 / 3^6 = 2
replace A1 with 2 in the second equation to get:
confirm by replacing A1 in the first eqution to get:
1458 = 2 * 3 ^ 6 = 1458.
do the same in the secone equation to get:
13122 = 2 * 3 ^ 8 = 13122.
you have solved for A1 and r.
A1 is 2 and r is 3.
if you want to find the second term algebraically, you would get:
A2 = 2 * 3^1 = 6
that's your solution.


you  could also have solved as follows:
A9 / A7 = (A1 * r^8)/A1 * r^6)
the A1 in the numerator and denominator cancel out and you are left with:
A9 / A7 = r^8 / r^6
simplify to get A9 / A7 = r^2
this becomes 13122 / 1458 = r^2
solve for r to get:
r = sqrt(13122 / 1458) = 3.
once you found r, then you can find A1 by just replacing r in the equation and solving for A1.
for example:
1458 = A1 * 3^6
this becomes 1458 = A1 * 749
solve for A1 to get:
A1 = 1458 / 729 = 2
you now have:
r = 3
A1 = 2
from there, you can easily find A2 because A2 = A1 * r ^ (2-1) which becomes A2 = 2 * 3 ^ 1 which becomes A2 = 2 * 3 which is equal to 6.