11 Determine the value(s) of $$ x $$ and/or $$ y $$ in each of the following geometric sequences: a $$ 2 ; x ; 32 ; \ldots $$ b $$ x+1 ; x-1 ; x ; \ldots $$ c $$ 20 ; x ; y ;-\frac{4}{25} $$ d $$ 7 x+1 ; 2 x+2 ; x-1 ; \ldots $$ e $$ 7 ; x ; y ; 56 ; \ldots $$

Question

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Let's solve each part step by step, identifying the values of $$ x $$ and/or $$ y $$ that make each sequence geometric. In a geometric sequence, the ratio between consecutive terms is constant.

### Problem 11a:
**Sequence:** $$ 2 ; x ; 32 ; \ldots $$

**Solution:**
For a geometric sequence, the ratio between consecutive terms is constant:
$$
\frac{x}{2} = \frac{32}{x}
$$
Cross-multiply to solve for $$ x $$:
$$
x^2 = 2 \times 32 = 64
$$
$$
x = \pm \sqrt{64} = \pm 8
$$

**Answer:** $$ x = 8 $$ or $$ x = -8 $$

---

### Problem 11b:
**Sequence:** $$ x+1 ; x-1 ; x ; \ldots $$

**Solution:**
Let the common ratio be $$ r $$. Then:
$$
\frac{x - 1}{x + 1} = \frac{x}{x - 1}
$$
Cross-multiply:
$$
(x - 1)^2 = x(x + 1)
$$
Expand both sides:
$$
x^2 - 2x + 1 = x^2 + x
$$
Subtract $$ x^2 $$ from both sides:
$$
-2x + 1 = x
$$
$$
1 = 3x
$$
$$
x = \frac{1}{3}
$$

**Answer:** $$ x = \dfrac{1}{3} $$

---

### Problem 11c:
**Sequence:** $$ 20 ; x ; y ; -\dfrac{4}{25} $$

**Solution:**
Let the common ratio be $$ r $$.

$$
x = 20r
$$
$$
y = x \cdot r = 20r^2
$$
$$
-\frac{4}{25} = y \cdot r = 20r^3
$$
Solve for $$ r $$:
$$
20r^3 = -\frac{4}{25}
$$
$$
r^3 = -\frac{4}{25 \times 20} = -\frac{4}{500} = -\frac{1}{125}
$$
$$
r = -\frac{1}{5}
$$

Now find $$ x $$ and $$ y $$:
$$
x = 20 \times \left(-\frac{1}{5}\right) = -4
$$
$$
y = 20 \times \left(-\frac{1}{5}\right)^2 = 20 \times \frac{1}{25} = \frac{20}{25} = \frac{4}{5}
$$

**Answer:** $$ x = -4 $$ and $$ y = \dfrac{4}{5} $$

---

### Problem 11d:
**Sequence:** $$ 7x + 1 ; 2x + 2 ; x - 1 ; \ldots $$

**Solution:**
Let the common ratio be $$ r $$.

$$
\frac{2x + 2}{7x + 1} = \frac{x - 1}{2x + 2}
$$
Cross-multiply:
$$
(2x + 2)^2 = (7x + 1)(x - 1)
$$
Expand both sides:
$$
4x^2 + 8x + 4 = 7x^2 - 7x + x - 1
$$
Simplify:
$$
4x^2 + 8x + 4 = 7x^2 - 6x - 1
$$
Bring all terms to one side:
$$
0 = 7x^2 - 6x - 1 - 4x^2 - 8x - 4
$$
$$
0 = 3x^2 - 14x - 5
$$
Solve the quadratic equation:
$$
x = \frac{14 \pm \sqrt{(-14)^2 - 4 \times 3 \times (-5)}}{2 \times 3} = \frac{14 \pm \sqrt{196 + 60}}{6} = \frac{14 \pm \sqrt{256}}{6} = \frac{14 \pm 16}{6}
$$
$$
x = \frac{30}{6} = 5 \quad \text{or} \quad x = \frac{-2}{6} = -\frac{1}{3}
$$

We need to verify which solution(s) make sense in the context of a geometric sequence (i.e., the ratios are consistent).

**For $$ x = 5 $$:**
$$
7(5) + 1 = 36; \quad 2(5) + 2 = 12; \quad 5 - 1 = 4
$$
Ratios:
$$
\frac{12}{36} = \frac{1}{3}; \quad \frac{4}{12} = \frac{1}{3}
$$
Consistent.

**For $$ x = -\frac{1}{3} $$:**
$$
7\left(-\frac{1}{3}\right) + 1 = -\frac{7}{3} + 1 = -\frac{4}{3}; \quad 2\left(-\frac{1}{3}\right) + 2 = -\frac{2}{3} + 2 = \frac{4}{3}; \quad -\frac{1}{3} - 1 = -\frac{4}{3}
$$
Ratios:
$$
\frac{\frac{4}{3}}{-\frac{4}{3}} = -1; \quad \frac{-\frac{4}{3}}{\frac{4}{3}} = -1
$$
Consistent.

**Answer:** $$ x = 5 $$ or $$ x = -\dfrac{1}{3} $$

---

### Problem 11e:
**Sequence:** $$ 7 ; x ; y ; 56 ; \ldots $$

**Solution:**
Let the common ratio be $$ r $$.

$$
x = 7r
$$
$$
y = x \cdot r = 7r^2
$$
$$
56 = y \cdot r = 7r^3
$$
Solve for $$ r $$:
$$
7r^3 = 56
$$
$$
r^3 = 8
$$
$$
r = 2
$$

Now find $$ x $$ and $$ y $$:
$$
x = 7 \times 2 = 14
$$
$$
y = 7 \times 2^2 = 7 \times 4 = 28
$$

**Answer:** $$ x = 14 $$ and $$ y = 28 $$

---

### Summary of Answers:
a) $$ x = 8 $$ or $$ x = -8 $$

---
a) $$ x = 8 $$ or $$ x = -8 $$ b) $$ x = \dfrac{1}{3} $$ c) $$ x = -4 $$ and $$ y = \dfrac{4}{5} $$ d) $$ x = 5 $$ or $$ x = -\dfrac{1}{3} $$ e) $$ x = 14 $$ and $$ y = 28 $$

11 Determine the value(s) of and/or in each of the following
geometric sequences:
a
b
c
d
e

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Solución

Problem 11a:

Problem 11b:

Problem 11c:

Problem 11d:

Problem 11e:

Summary of Answers:

Beyond the Answer

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11 Determine the value(s) of and/or in each of the following geometric sequences: a b c d e

Solución de inteligencia artificial de Upstudy

Responder

Solución

Problem 11a:

Problem 11b:

Problem 11c:

Problem 11d:

Problem 11e:

Summary of Answers:

Beyond the Answer

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11 Determine the value(s) of and/or in each of the following
geometric sequences:
a
b
c
d
e