The number of terms of a geometric sequence is even.
- The sum of the odd terms, that is - terms in the odd places (First term + Third Term + Fifth Term + ... and so on) is 4329
- The sum of the even terms, that is – terms in the even places (Second Term + Fourth Term + Sixth Term + ... and so on) is 5772
- The last term exceeds the first by 2343
Determine the geometric sequence.
Let a = first term; r = common ration; and 2n = number of terms.
For #1, we have a geometric series with first term = a, common ration = r^2 and number of terms = n.
For #2, we have a geometric series with first term = ar, common ration = r^2 and number of terms = n.
Apply the standard formula for the sum of a geometric series to each of these and divide one by the other: r = 5771/4329 = 4/3.
A bit of algebraic manipulation, after substituting this value for r into each of the 2 equations for the geometric sums, yields n = 3.
Therefore, the original series has 2n = 6 terms.
Substituting these values for r and n into the sum of the odd terms gives a = 729.
So, the original series is: 729; 729*(4/3) = 972; 1,296; 1,728; 2,304; 3,072.