Sequences and series
GEOMETRIC SEQUENCES
is a geometric sequence and
is an arithmetic sequence.
Find the values of and .

Ask by Hanson Lewis. in South Africa

Mar 28,2025

Question

Sequences and series GEOMETRIC SEQUENCES $$ -\frac{1}{2} ; p ; y $$ is a geometric sequence and $$ p ; y ;-\frac{1}{2} $$ is an arithmetic sequence. Find the values of $$ p $$ and $$ y $$.

UpStudy AI · Accepted Answer

1. For a geometric sequence, the square of the middle term equals the product of the first and third terms. Thus, for the sequence
   $$
   -\frac{1}{2}, \; p, \; y,
   $$
   we have:
   $$
   p^2 = \left(-\frac{1}{2}\right)y.
   $$
   Rearranging, we obtain:
   $$
   y = -2p^2.
   $$

2. For an arithmetic sequence, the middle term is the average of the first and third terms. Thus, for the sequence
   $$
   p, \; y, \; -\frac{1}{2},
   $$
   we have:
   $$
   y = \frac{p + \left(-\frac{1}{2}\right)}{2} = \frac{p - \frac{1}{2}}{2}.
   $$

3. Setting the two expressions for $$ y $$ equal:
   $$
   -2p^2 = \frac{p - \frac{1}{2}}{2}.
   $$
   Multiply both sides by 2 to eliminate the fraction:
   $$
   -4p^2 = p - \frac{1}{2}.
   $$
   Rearranging gives:
   $$
   -4p^2 - p + \frac{1}{2} = 0.
   $$
   Multiply the entire equation by 2 to clear the fraction:
   $$
   -8p^2 - 2p + 1 = 0.
   $$
   Multiply by $$-1$$ to simplify:
   $$
   8p^2 + 2p - 1 = 0.
   $$

4. Solve the quadratic equation using the quadratic formula:
   $$
   p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
   $$
   with $$a = 8$$, $$b = 2$$, and $$c = -1$$:
   $$
   p = \frac{-2 \pm \sqrt{2^2 - 4(8)(-1)}}{2 \cdot 8} = \frac{-2 \pm \sqrt{4 + 32}}{16} = \frac{-2 \pm \sqrt{36}}{16}.
   $$
   Thus:
   $$
   p = \frac{-2 \pm 6}{16}.
   $$
   
   This gives the two solutions:
   - $$ p = \frac{-2 + 6}{16} = \frac{4}{16} = \frac{1}{4} $$.
   - $$ p = \frac{-2 - 6}{16} = \frac{-8}{16} = -\frac{1}{2} $$.

5. Find $$ y $$ using $$ y = -2p^2 $$:
   - If $$ p = \frac{1}{4} $$:
     $$
     y = -2\left(\frac{1}{4}\right)^2 = -2\left(\frac{1}{16}\right) = -\frac{1}{8}.
     $$
   - If $$ p = -\frac{1}{2} $$:
     $$
     y = -2\left(-\frac{1}{2}\right)^2 = -2\left(\frac{1}{4}\right) = -\frac{1}{2}.
     $$

6. Final answers:
   - Either $$ p = \frac{1}{4} $$ and $$ y = -\frac{1}{8} $$, 
   - or $$ p = -\frac{1}{2} $$ and $$ y = -\frac{1}{2} $$.
$$ p = \frac{1}{4} $$ and $$ y = -\frac{1}{8} $$, or $$ p = -\frac{1}{2} $$ and $$ y = -\frac{1}{2} $$.

Sequences and series
GEOMETRIC SEQUENCES
is a geometric sequence and
is an arithmetic sequence.
Find the values of and .

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Sequences and series GEOMETRIC SEQUENCES is a geometric sequence and is an arithmetic sequence. Find the values of and .