the sum of the first six terms of a G. P is 9 times the sum of the first three terms . what is the first term ?

Without any more information, the first term can be any number.

Let a be the first term in the geometric progression, and let r be the common ratio between consecutive terms. Then

a + ar + ar ² + ar ³ + ar ⁴ + ar ⁵ = 9 (a + ar + ar ²)

ar ³ + ar ⁴ + ar ⁵ = 8 (a + ar + ar ²)

r ³ + r ⁴ + r ⁵ = 8 (1 + r + r ²)

r ³ (1 + r + r ²) = 8 (1 + r + r ²)

r ³ = 8

r = 2

Now with this ratio, the sum of the first six terms is

a (1 + r + r ² + r ³ + r ⁴ + r ⁵) = a (1 - r ⁶)/(1 - r) = 63a

while the sum of the first three terms is

a (1 + r + r ²) = a (1 - r ³)/(1 - r) = 7a

and of course 63a = 9 • 7a, and this is true for any a.

eve and marks ratios are equivalent

step-by-step explanation:

check above link.

graphically?   well, you graph them both, the solutions is where they both intersect each other, since there are two solutions, meaning they will meet twice, or namely the linear will run into the quadratic curve twice, and where they meet, those x,y coordinates are the solutions to the system.