Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find the 3 rd term in the expansion of
(3x-2)^(4) in simplest form.
Answer:

Find the $3$ rd term in the expansion of $(3 x-2)^{4}$ in simplest form. $\newline$ Answer:

View step-by-step help

View step-by-step help

Partial sums of geometric series

Full solution

Q. Find the

3

rd term in the expansion of

(3 x-2)^{4}

in simplest form.

\newline

Answer:

Use Binomial Theorem: To find the $3$ rd term in the expansion of $(3x-2)^{4}$ , we will use the binomial theorem. The binomial theorem states that the $n$ th term in the expansion of $(a+b)^{m}$ is given by the formula: $T(n) = C(m, n-1) \cdot a^{(m-n+1)} \cdot b^{(n-1)}$ , where $C(m, n-1)$ is the binomial coefficient. For the $3$ rd term, $n=3$ .
Calculate Binomial Coefficient: First, we calculate the binomial coefficient for the $3$ rd term, which is $C(4, 3-1)$ or $C(4, 2)$ . The binomial coefficient $C(4, 2)$ is calculated as $\frac{4!}{2! \cdot (4-2)!}$ , which simplifies to $\frac{4\cdot3}{2\cdot1} = 6$ .
Calculate Powers of a and b: Next, we calculate the powers of $a$ and $b$ for the $3$ rd term. Since $a$ is $3x$ and $b$ is $-2$ , and we are looking for the $3$ rd term ( $n=3$ ), we have $a^{4-3+1} = (3x)^{2}$ and $b$ $0$ .
Multiply Coefficient and Powers: Now we multiply the binomial coefficient by the powers of $a$ and $b$ . So the $3$ rd term is $6 \times (3x)^{2} \times (-2)^{1}$ . This simplifies to $6 \times 9x^2 \times -2$ , which is $-108x^2$ .
Final Result: Therefore, the $3^{\text{rd}}$ term in the expansion of $(3x-2)^{4}$ in simplest form is $-108x^2$ .

More problems from Partial sums of geometric series

Question

Find the sum of the finite arithmetic series.

\sum_{n=1}^{10} (7n+4)

\newline

Posted 2 months ago

Question

Type the missing number in this sequence: `55,\59,\63,\text{_____},\71,\75,\79`

Posted 1 month ago

Question

Type the missing number in this sequence: `2, 4 8`, ______,`32`

Posted 2 months ago

Question

What kind of sequence is this?

2, 10, 50, 250, \ldots

Choices:Choices:

\newline

\text{[A]arithmetic}

\newline

\text{[B]geometric}

\newline

\text{[C]both}

\newline

\text{[D]neither}

Posted 2 months ago

Question

What is the missing number in this pattern?

1, 4, 9, 16, 25, 36, 49, 64, 81, \_\_\_\_

Posted 1 month ago

Question

Classify the series.

\sum_{n = 0}^{12} (n + 2)^3

\newline

\newline

\text{[A]arithmetic}

\newline

\text{[B]geometric}

\newline

\text{[C]both}

\newline

\text{[D]neither}

Posted 1 month ago

Question

Find the first three partial sums of the series.

\newline

1 + 6 + 11 + 16 + 21 + 26 + \cdots

\newline

Write your answers as integers or fractions in simplest form.

\newline

S_1 =

\newline

S_2 =

\newline

S_3 =

Posted 1 month ago

Question

Find the third partial sum of the series.

\newline

3 + 9 + 15 + 21 + 27 + 33 + \cdots

\newline

Write your answer as an integer or a fraction in simplest form.

\newline

S_3 =

Posted 1 month ago

Question

Find the first three partial sums of the series.

\newline

1 + 7 + 13 + 19 + 25 + 31 + \cdots

\newline

Write your answers as integers or fractions in simplest form.

\newline

S_1 =

\newline

S_2 =

\newline

S_3 =

Posted 2 months ago

Question

Does the infinite geometric series converge or diverge?

\newline

1 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \cdots

\newline

\newline

\text{[A]converge}

\newline

\text{[B]diverge}

Posted 1 month ago