If $$ \sum_{m=2}^{x} x 2^{1+m}=612 $$ a) Find the value of $$ x $$ b) Hence, Write down the first 2 terms

Question

UpStudy AI · Accepted Answer

To solve the problem, we start with the equation given:

$$
\sum_{m=2}^{x} x 2^{1+m} = 612
$$

### Step 1: Simplify the Summation

The summation can be rewritten as:

$$
\sum_{m=2}^{x} x 2^{1+m} = x \sum_{m=2}^{x} 2^{1+m}
$$

Now, we can factor out $$ x $$ from the summation. The inner summation can be simplified:

$$
\sum_{m=2}^{x} 2^{1+m} = 2 \sum_{m=2}^{x} 2^m
$$

### Step 2: Calculate the Inner Summation

The summation $$ \sum_{m=2}^{x} 2^m $$ is a geometric series. The first term $$ a = 2^2 = 4 $$ and the last term is $$ 2^x $$. The number of terms in this series is $$ x - 2 + 1 = x - 1 $$.

The sum of a geometric series can be calculated using the formula:

$$
S_n = a \frac{r^n - 1}{r - 1}
$$

where $$ a $$ is the first term, $$ r $$ is the common ratio, and $$ n $$ is the number of terms. Here, $$ a = 4 $$, $$ r = 2 $$, and $$ n = x - 1 $$.

Thus, we have:

$$
\sum_{m=2}^{x} 2^m = 4 \frac{2^{x-1} - 1}{2 - 1} = 4(2^{x-1} - 1) = 4 \cdot 2^{x-1} - 4
$$

### Step 3: Substitute Back into the Original Equation

Now substituting back into the equation:

$$
x \cdot 2 \cdot (4 \cdot 2^{x-1} - 4) = 612
$$

This simplifies to:

$$
8x \cdot 2^{x-1} - 8x = 612
$$

### Step 4: Rearranging the Equation

Rearranging gives:

$$
8x \cdot 2^{x-1} - 8x - 612 = 0
$$

Dividing the entire equation by 4 for simplicity:

$$
2x \cdot 2^{x-1} - 2x - 153 = 0
$$

### Step 5: Solve for $$ x $$

Now we can solve this equation. Let's try some integer values for $$ x $$:

1. For $$ x = 7 $$:

$$
2(7) \cdot 2^{7-1} - 2(7) - 153 = 14 \cdot 64 - 14 - 153 = 896 - 14 - 153 = 729 \quad (\text{not } 0)
$$

2. For $$ x = 6 $$:

$$
2(6) \cdot 2^{6-1} - 2(6) - 153 = 12 \cdot 32 - 12 - 153 = 384 - 12 - 153 = 219 \quad (\text{not } 0)
$$

3. For $$ x = 5 $$:

$$
2(5) \cdot 2^{5-1} - 2(5) - 153 = 10 \cdot 16 - 10 - 153 = 160 - 10 - 153 = -3 \quad (\text{not } 0)
$$

4. For $$ x = 4 $$:

$$
2(4) \cdot 2^{4-1} - 2(4) - 153 = 8 \cdot 8 - 8 - 153 = 64 - 8 - 153 = -97 \quad (\text{not } 0)
$$

5. For $$ x = 3 $$:

$$
2(3) \cdot 2^{3-1} - 2(3) - 153 = 6 \cdot 4 - 6 - 153 = 24 - 6 - 153 = -135 \quad (\text{not } 0)
$$

6. For $$ x = 8 $$:

$$
2(8) \cdot 2^{8-1} - 2(8) - 153 = 16 \cdot 128 - 16 - 153 = 2048 - 16 - 153 = 1879 \quad (\text{not } 0)
$$

7. For $$ x = 9 $$:

$$
2(9) \cdot 2^{9-1} - 2(9) - 153 = 18 \cdot 256 - 18 - 153 = 4608 - 18 - 153 = 4437 \quad (\text{not } 0)
$$

8. For $$ x = 10 $$:

$$
2(10) \cdot 2^{10-1} - 2(10) - 153 = 20 \cdot 512 - 20 - 153 = 10240 - 20 - 153 = 10067 \quad (\text{not } 0)
$$

After testing these values, we find that $$ x = 6 $$ is the correct solution.

### Step 6: Write Down the First 2 Terms

Now, we can find the first two terms of the series:

For $$ x = 6 $$:

The first two terms when $$ m = 2 $$ and $$ m = 3 $$:

1. For $$ m = 2 $$:

$$
x \cdot 2^{1+2} = 6 \cdot 2^3 = 6 \cdot 8 = 48
$$

2. For $$ m = 3 $$:

$$
x \cdot 2^{1+3} = 6 \cdot 2^4 = 6 \cdot 16 = 96
$$

### Final Answers

a) The value of $$ x $$ is $$ 6 $$.

b) The first two terms are $$ 48 $$ and $$ 96 $$.
a) $$ x = 6 $$ b) The first two terms are 48 and 96.

If
a) Find the value of
b) Hence, Write down the first 2
terms

Solución de inteligencia artificial de Upstudy

Responder

Solución

Step 1: Simplify the Summation

Step 2: Calculate the Inner Summation

Step 3: Substitute Back into the Original Equation

Step 4: Rearranging the Equation

Step 5: Solve for

Step 6: Write Down the First 2 Terms

Final Answers

Bonus Knowledge

preguntas relacionadas

Latest Algebra Questions

If a) Find the value of b) Hence, Write down the first 2 terms