To solve this problem, we need to use the formula for the nth term of a geometric sequence:
An = a*r^(n-1)
where An is the nth term, a is the first term, r is the common ratio, and n is the position of the term in the sequence.
a) To determine the common ratio "r," we can use the information provided about the sum of the 2nd and 5th terms:
A2 + A5 = a*r + a*r^4 = 13
Similarly, the sum of the 3rd and 6th terms is:
A3 + A6 = a*r^2 + a*r^5 = -39
Now we have obtained two equations:
1) a*r + a*r^4 = 13
2) a*r^2 + a*r^5 = -39
To solve these equations simultaneously, we can use algebraic manipulation or substitution. Let's solve this system of equations by substitution:
Rearranging equation 1) to express a in terms of r:
a = 13 / (r + r^4)
Substituting this value of a into equation 2):
(13 / (r + r^4)) * r^2 + (13 / (r + r^4)) * r^5 = -39
Now we have a single equation in terms of r. We can multiply the equation throughout by (r + r^4) to simplify:
13r^2 + 13r^5 = -39(r + r^4)
Simplifying further:
13r^2 + 13r^5 = -39r - 39r^4
Rearranging and factoring:
13r^2 + 39r - 39r^4 - 13r^5 = 0
Now we have a polynomial equation in terms of r. To solve this equation, we can factor out the common terms:
r(13r + 39) - r^4(39r + 13) = 0
Factoring out r gives:
r((13r + 39) - r^3(39r + 13)) = 0
From this equation, we can see two possible solutions:
1) r = 0
2) 13r + 39 = r^3(39r + 13)
The solution r = 0 does not yield a valid geometric sequence since the common ratio cannot be zero. Therefore, we need to solve the second equation to find the value of r.
b) To determine the first term "a" of the sequence, we can substitute the value of r obtained from the previous step into one of the original equations (either equation 1) or 2)). Let's use equation 1) as an example:
a * r + a * r^4 = 13
Substituting r = 0 into this equation would also give us a = 0, which is not a valid solution. Therefore, we use the equation derived from equation 2):
a * r^2 + a * r^5 = -39
Substituting the value of r obtained from the previous step into this equation will give us the value of a.
Note: The second equation may have multiple valid solutions. To find the exact value of a, we need to substitute the value of r into the equation and then solve for a.