Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find the 4 th term in the expansion of
(2x-3)^(6) in simplest form.
Answer:

Find the $4$ th term in the expansion of $(2 x-3)^{6}$ 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

4

th term in the expansion of

(2 x-3)^{6}

in simplest form.

\newline

Answer:

Identify Variables: To find the $4$ th term in the expansion of $(2x-3)^{6}$ , 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 $C(m, n-1) \cdot a^{m-n+1} \cdot b^{n-1}$ , where $C(m, n-1)$ is the binomial coefficient, which can be calculated as $\frac{m!}{(m-n+1)! \cdot (n-1)!}$ .
Calculate Binomial Coefficient: First, we need to identify $a$ , $b$ , and $m$ in our expression $(2x-3)^{6}$ . Here, $a = 2x$ , $b = -3$ , and $m = 6$ .
Calculate $a^{(m-n+1)}$ : Now, we calculate the $4^{th}$ term, which means $n = 4$ . We need to find $C(6, 4-1)$ which is $C(6, 3)$ . The binomial coefficient $C(6, 3)$ is calculated as $6! / [3! * (6-3)!] = (6*5*4) / (3*2*1) = 20$ .
Calculate $b^{(n-1)}$ : Next, we calculate $a^{(m-n+1)}$ which is $(2x)^{(6-4+1)} = (2x)^{3}$ . This equals $8x^3$ .
Multiply Coefficients: Then, we calculate $b^{(n-1)}$ which is $(-3)^{(4-1)} = (-3)^{3} = -27$ .
Final Result: Now, we multiply the binomial coefficient by the powers of 'a' and 'b' to get the $4$ th term: $20 \times 8x^3 \times (-27)$ . This simplifies to $20 \times -216x^3$ .
Final Result: Now, we multiply the binomial coefficient by the powers of 'a' and 'b' to get the $4$ th term: $20 \times 8x^3 \times (-27)$ . This simplifies to $20 \times -216x^3$ . Finally, we multiply $20$ by $-216$ to get the coefficient of $x^3$ in the $4$ th term: $20 \times -216 = -4320$ . So, the $4$ th term is $-4320x^3$ .

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 9 months ago

Question

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

Posted 5 months ago

Question

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

Posted 5 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 5 months ago

Question

What is the missing number in this pattern?

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

Posted 7 months 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 5 months 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 7 months 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 7 months 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 9 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 5 months ago

Related topics

Algebra - Order of Operations Algebra - Distributive Property `X` and `Y` Axes Geometry - Scalene Triangle Common Multiple Geometry - Quadrant