Find the 4 th term of $$ (x+5)^{7} $$ $$ \frac{625 x^{4}}{4375 x^{4}} $$ $$ \frac{21,875 x^{3}}{35 x^{4}} $$

Question

UpStudy AI · Accepted Answer

To find the 4th term of the expression $$ (x+5)^{7} $$, we can use the binomial theorem. The general formula for the $$ k $$th term of $$ (a+b)^{n} $$ is given by:

$$ T_{k+1} = \binom{n}{k} \cdot a^{n-k} \cdot b^{k} $$

In this case, $$ a = x $$, $$ b = 5 $$, and $$ n = 7 $$. We want to find the 4th term, which corresponds to $$ k = 3 $$ in the formula.

Substitute the values into the formula:

$$ T_{4} = \binom{7}{3} \cdot x^{7-3} \cdot 5^{3} $$

$$ T_{4} = \binom{7}{3} \cdot x^{4} \cdot 5^{3} $$

Now, we can calculate the value of $$ \binom{7}{3} $$ and simplify the expression to find the 4th term.
Calculate the value by following steps:
- step0: Calculate:
  $$ { }_{7}C_{3}$$
- step1: Expand the expression:
  $$\frac{7!}{3!\times \left(7-3\right)!}$$
- step2: Subtract the numbers:
  $$\frac{7!}{3!\times 4!}$$
- step3: Expand the expression:
  $$\frac{7\times 6\times 5\times 4\times 3!}{3!\times 4!}$$
- step4: Simplify:
  $$\frac{7\times 6\times 5\times 4}{4!}$$
- step5: Multiply the terms:
  $$\frac{840}{4!}$$
- step6: Calculate the value:
  $$\frac{840}{24}$$
- step7: Reduce the fraction:
  $$35$$
The value of $$ \binom{7}{3} $$ is 35.

Now, substitute this value back into the expression for the 4th term:

$$ T_{4} = 35 \cdot x^{4} \cdot 5^{3} $$

$$ T_{4} = 35 \cdot x^{4} \cdot 125 $$

$$ T_{4} = 4375x^{4} $$

Therefore, the 4th term of $$ (x+5)^{7} $$ is $$ 4375x^{4} $$.
The 4th term of $$ (x+5)^{7} $$ is $$ 4375x^{4} $$.

Find the 4 th term of

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Algebra Questions