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
(x+5)^(7) in simplest form.
Answer:

Find the $3$ rd term in the expansion of $(x+5)^{7}$ 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

(x+5)^{7}

in simplest form.

\newline

Answer:

Use Binomial Theorem: To find the $3^{\text{rd}}$ term in the expansion of $(x+5)^{7}$ , we will use the binomial theorem. The binomial theorem states that the $n^{\text{th}}$ term in the expansion of $(a+b)^{m}$ is given by the formula $C(m, k-1) \cdot a^{m-k+1} \cdot b^{k-1}$ , where $C(m, k-1)$ is the binomial coefficient, which can be calculated as $\frac{m!}{(m-k+1)! \cdot (k-1)!}$ . In this case, the $3^{\text{rd}}$ term corresponds to $k=3$ .
Calculate Binomial Coefficient: First, we calculate the binomial coefficient for the $3$ rd term, which is $C(7, 3-1) = C(7, 2)$ . This can be calculated as $\frac{7!}{(7-2)! \times 2!} = \frac{7!}{5! \times 2!} = \frac{7 \times 6}{2 \times 1} = 21$ .
Apply Binomial Theorem: Next, we apply the binomial theorem formula to find the $3$ rd term. We have $a = x$ , $b = 5$ , $m = 7$ , and $k = 3$ . So the $3$ rd term is $C(7, 2) \times x^{7-2} \times 5^{2} = 21 \times x^{5} \times 25$ .
Simplify Expression: Now, we simplify the expression. Multiplying the binomial coefficient by the powers of $x$ and $5$ , we get $21 \times x^{5} \times 25 = 525 \times x^{5}$ .

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