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Step-by-step Solution
Step 1: Understand the Given Equation for AM and GM
We are told that the arithmetic mean (AM) and geometric mean (GM) of the
p-th and q-th terms of the sequence satisfy the equation
4x^2 - 9x + 5 = 0 .
Solving the quadratic equation 4x^2 - 9x + 5 = 0 :
Factorize to get:
(x - 1)(4x - 5) = 0
So, x = 1 or x = \frac{5}{4} .
Because AM ≥ GM for any two non-negative real numbers, we take:
AM = \frac{5}{4} and GM = 1 .
Step 2: Express the General Term of the Given Sequence
The sequence given is: –16, 8, –4, 2, …
Notice that each term is obtained by multiplying the previous term
by the common ratio -\frac{1}{2} . Therefore, this is a geometric
progression (GP) with first term a = -16 and common ratio r = -\frac{1}{2} .
The general term t_n of a GP is given by:
t_n = a \, r^{n-1} .
So, for our sequence:
t_n = -16 \left(-\frac{1}{2}\right)^{n-1} .
Step 3: Write Down the p-th and q-th Terms
The p-th term is:
t_p = -16\left(-\frac{1}{2}\right)^{p-1} .
The q-th term is:
t_q = -16\left(-\frac{1}{2}\right)^{q-1} .
Step 4: Apply the AM Condition
We know the arithmetic mean (AM) of t_p and t_q is \frac{5}{4} .
By definition of AM:
\displaystyle \text{AM} = \frac{t_p + t_q}{2} = \frac{5}{4} .
Hence,
t_p + t_q = \frac{5}{2} .
Step 5: Apply the GM Condition
The geometric mean (GM) of t_p and t_q is given as 1. Therefore:
\sqrt{t_p \, t_q} = 1 \quad \Rightarrow \quad t_p \, t_q = 1 .
Step 6: Simplify the Expression for t_p \, t_q = 1
Substituting the expressions for t_p and t_q :
t_p \, t_q = \bigl[-16\left(-\frac{1}{2}\right)^{p-1}\bigr]\cdot
\bigl[-16\left(-\frac{1}{2}\right)^{q-1}\bigr]
= 1 .
This simplifies to
(-16)(-16)\left(-\frac{1}{2}\right)^{(p-1)+(q-1)}
= 1 .
Thus,
256\left(-\frac{1}{2}\right)^{p+q-2} = 1 .
Step 7: Solve for p + q
Rewrite 256 as (-2)^8 (since 256 = 2^8 , but we note the alternating sign
handled by (-1)^8 = 1):
( -2 )^8 \left(-\frac{1}{2}\right)^{p+q-2} = 1 .
Notice that (-2)^8 = 256 , and to equate it to 1, we must have
the exponent match so that the product simplifies properly.
By equating powers, we find
p + q = 10 .
Final Answer
The value of p + q is 10.
