The first three term of a geometric sequence are T1;T2 and T3. If T2=T1+4 and T3=T2 +9, determine the values of T1;T2 and T3
ar = a+4
ar^2 = ar+9
r = 9/4
a = 16/5
so, now figure the first 3 terms.
let the first 3 terms be
a , ar , and ar^2
ar = a + 4
ar - a = 4
a(r-1) = 4
a = 4/(r-1)
ar^2 = ar + 9
ar^2 - ar = 9
a = (9/(r^2 - r)
so 9/(r^2-r) = 4/(r-1)
4r^2 - 4r = 9r - 9
4r^2 - 13r + 9 = 0
(4r - 9)(r - 1) = 0
r = 9/4 or r = 1
if r = 9/4,
a = 4/(9/4 - 1) = 16/5
the first 3 terms are 16/5 , 36/5 , and 81/5
if r = 1 , a = 4/0 , which would be undefined, so r≠1
a = 9/4, r = 16/5 yields the only solution
check:
first condition: t2 = t1 + 4
LS = 36/5
RS = 16/5 + 4 = 36/5 , check
2nd condition: t3 = t2 + 9
LS = 81/5
RS =36/5 + 9 =81/5 , check:
To determine the values of T1, T2, and T3, we can use the information given about the geometric sequence.
1. Start by using the equation for a geometric sequence: Tn = T1 * r^(n-1), where Tn represents the n-th term, T1 is the first term, r is the common ratio, and n is the position of the term.
2. Given that T2 = T1 + 4, we can substitute these values into the equation:
T2 = T1 * r^(2-1) = T1 * r = T1 + 4
3. Similarly, for T3 = T2 + 9, we can substitute these values into the equation:
T3 = T1 * r^(3-1) = T1 * r^2 = T2 + 9 = (T1 + 4) + 9
4. Now we have a system of equations with T1 and r:
T1 * r = T1 + 4 (Equation 1)
T1 * r^2 = (T1 + 4) + 9 (Equation 2)
5. Simplify Equation 2:
T1 * r^2 = T1 + 13 (Equation 2)
6. We can rewrite Equation 1 as follows:
T1 * r - T1 = 4
7. Rearrange the above equation to isolate T1:
T1 * (r - 1) = 4
T1 = 4 / (r - 1)
8. Substitute this value of T1 into Equation 2:
(4/(r-1)) * r^2 = (4/(r-1)) + 13
9. Multiply through by (r-1):
4 * r^2 = 4 + 13 * (r - 1)
10. Expand and rearrange the equation:
4 * r^2 = 4 + 13r - 13
4 * r^2 - 13r - 9 = 0
11. This is now a quadratic equation in terms of r. You can solve it using the quadratic formula or factoring. Once you find the value(s) of r, substitute it back into Equation 1 to find T1.
12. Lastly, once you have T1 and r, use the geometric sequence formula to find T2 and T3:
T2 = T1 * r
T3 = T2 * r
By following these steps, you should be able to determine the values of T1, T2, and T3 for the given geometric sequence.