hunter uses a running total to keep track of his score in a card game, and he notices that his score follows an arithmetic sequence. which of these sequences could be hunters scores?
show your work here
$\boldsymbol{\circ}$ $-1, -8, -27, -64, -125, \dots$ $\boldsymbol{\circ}$ $-5, -2, 3, 10, 19, \dots$
$\boldsymbol{\circ}$ $-5, -1, 3, 7, 11, \dots$ $\boldsymbol{\circ}$ $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \dots$
6. what is the solution of the system of equations?
$4x + 2y = -1$
$3x + 4y = 3$

Question

Accepted Answer

### Part 1: Arithmetic Sequence Identification
# Explanation:
## Step 1: Recall Arithmetic Sequence Definition
An arithmetic sequence has a common difference $ d $, where $ d = a_{n+1} - a_n $ for all $ n $.

## Step 2: Analyze First Sequence ($-1, -8, -27, -64, -125, \dots$)
Check differences:
$-8 - (-1) = -7$, $-27 - (-8) = -19$. Not constant. Not arithmetic.

## Step 3: Analyze Second Sequence ($-5, -2, 3, 10, 19, \dots$)
Differences:
$-2 - (-5) = 3$, $3 - (-2) = 5$. Not constant. Not arithmetic.

## Step 4: Analyze Third Sequence ($-5, -1, 3, 7, 11, \dots$)
Differences:
$-1 - (-5) = 4$, $3 - (-1) = 4$, $7 - 3 = 4$, $11 - 7 = 4$. Constant difference $ d = 4 $. Arithmetic.

## Step 5: Analyze Fourth Sequence ($\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \dots$)
Differences:
$\frac{1}{4} - \frac{1}{2} = -\frac{1}{4}$, $\frac{1}{8} - \frac{1}{4} = -\frac{1}{8}$. Not constant. Geometric (ratio $ \frac{1}{2} $). Not arithmetic.

# Answer (Arithmetic Sequence):
$-5, -1, 3, 7, 11, \dots$

### Part 2: System of Equations Solution
# Explanation:
## Step 1: Write Equations
Equation 1: $ 4x + 2y = -1 $
Equation 2: $ 3x + 4y = 3 $

## Step 2: Eliminate $ y $ (Multiply Eq1 by 2)
$ 8x + 4y = -2 $ (New Eq1)
$ 3x + 4y = 3 $ (Eq2)

## Step 3: Subtract Eq2 from New Eq1
$ (8x - 3x) + (4y - 4y) = -2 - 3 $
$ 5x = -5 $
$ x = -1 $

## Step 4: Substitute $ x = -1 $ into Eq1
$ 4(-1) + 2y = -1 $
$ -4 + 2y = -1 $
$ 2y = 3 $
$ y = \frac{3}{2} $

# Answer (System Solution):
$ x = -1 $, $ y = \frac{3}{2} $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Part 1: Arithmetic Sequence Identification

Step 1: Recall Arithmetic Sequence Definition

Step 2: Analyze First Sequence (\(-1, -8, -27, -64, -125, \dots\))

Step 3: Analyze Second Sequence (\(-5, -2, 3, 10, 19, \dots\))

Step 4: Analyze Third Sequence (\(-5, -1, 3, 7, 11, \dots\))

Step 5: Analyze Fourth Sequence (\(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \dots\))

Step 1: Write Equations

Step 2: Eliminate \( y \) (Multiply Eq1 by 2)

Step 3: Subtract Eq2 from New Eq1

Step 4: Substitute \( x = -1 \) into Eq1

Answer:

Part 2: System of Equations Solution