The 3rd and 6th term of a G.P are 108 and -32 respectively.Find the sum of the first 7th term.

a + a r + a r^2 + a r^3 + a r^4 + a r^5 + ar^6

a r^2 = 108
a r^5 =-32
r^5/r^2 = - 32/108 = -8/27
r^3 = -2^3/3^3 =
r = - 2/3
go back and get "a" now
then if you still need help summing go here:
https://www.mathsisfun.com/algebra/sequences-sums-geometric.html

S7 = a(1-r^7)/(1-r)

Agree

Third term = T3

Sixth term = T6
Tn = ar^n-1
T3=ar^2=108
ar^2=108_ _ _ _(1)
T6=ar^5=-32
ar^5=-32_ _ _ _(2)
divide eqn (2) by (1)
ar^5/ar^2=-32/108
r^3=-8/27
Cuberoot b.s
r=-2/3
Substitute r into eqn (1)
a(-2/3)^2=108
a×4/9=108
a=108×9/4=27×9=243
Sum of first 7 terms = S7
S7=a(1-r^n)/1-r
=243(1-(-2/3)/1-(-2/3)
=243(1+2/3)/1+2/3
=243(5/3)/5/3
=81×5×3/5=243
Sum of first 7 terms=243

Well, let's see here. We have a geometric progression (GP) and we know the values of the 3rd and 6th term. That's a good start, but we need a bit more information to find the sum of the first 7 terms. Any chance you know the common ratio between the terms? If not, we might need to do some detective work. Good thing I brought my magnifying glass and detective hat!

To find the sum of the first 7 terms of a geometric progression (G.P.), we first need to determine the common ratio (r) of the G.P. Given that the 3rd term is 108 and the 6th term is -32, we can use these values to find the common ratio.

Let's first find the value of the 4th term:
T4 = T3 * r
108 * r = T4

Similarly, we can find the value of the 5th term:
T5 = T4 * r
T5 = 108 * r * r = 108 * r^2

Now, let's find the value of the 6th term:
-32 = T5 * r
-32 = 108 * r^2 * r = 108 * r^3

We can now solve this cubic equation to determine the common ratio (r). There are different methods to solve cubic equations, such as factoring, synthetic division, or using a graphing calculator. Once you find the value of r, you can substitute it back into the equations to find the specific terms.

Once you have the common ratio and the first term (T1), you can use the formula for the sum of a G.P. to find the sum of the first 7 terms:

S7 = T1 * (r^7 - 1) / (r - 1)

Therefore, the sum of the first 7 terms can be obtained by substituting the values of T1 and r into the formula.