Find the 6th and 8th term of the geometric progression if their sum is 14 and the product

Find the 6th and 8th term of the geometric progression if their sum is 14 and the product of the 10th and 4th terms of this progression is 48.

1. For a given geometric progression B (n), the ratios of its members are known:

B6 + B8 = 14;

B4 * B10 = 48;

2. Let’s write the sum of terms in canonical form:

B6 + B8 = (B1 * q ^ 5) + (B1 * q ^ 7) = 14;

B6 = 14 – B8;

3. Artwork:

(B1 * q³) * (B1 * q ^ 9) = B1 * B1 * q³ * Q ^ 9 =

B1 * B1 * q³ * (q² * q ^ 7) = (B1 * q ^ 5) * (B1 * q ^ 7) = B6 * B8 = 48;

4. Calculate:

B6 * B8 = (14 – B8) * B8 = 14 * B8 – B8² = 48;

B8² – 14 * B * + 48 = 0;

B81.2 = 7 + – sqrt (7² – 48) = 7 + – 1;

B81 = 7 – 1 = 6;

B61 = 14 – 6 = 8;

B82 = 7 + 1 = 8;

B62 = 14 – 8 = 6.

Answer: the sixth term of the progression B (n) is 6, the eighth is 8.