Question 1094991
Can someone help me to solve this?
When (1-3x)^n is expanded, the coefficient of x^2 is 90. How to find n algebraically?
<pre>This is saying that there’s a specific term that’s 90x<sup>2</sup>

The formula to find a SPECIFIC term in a BINOMIAL expansion is: 
{{{matrix(1,3, (a + b)^n, "=", " " [n]C[r - 1]*(a)^(n - r + 1)*(b)^(r - 1))}}}, where r = term number 
{{{matrix(1,3, (1 - 3x)^n, "=", " " [n]C[r - 1]*(1)^(n - r + 1)*(- 3x)^(r - 1))}}} ------ Substituting 1 for a, and - 3x for b
{{{matrix(1,3, (1 - 3x)^n, "=", " " [n]C[r - 1] * 1 * (- 3x)^(r - 1))}}} ------- Replacing {{{(1)^(n - r + 1)}}} with 1 as 1 raised to ANY POWER equals 1  
{{{matrix(1,3, (1 - 3x)^n, "=", " " [n]C[r - 1] * (- 3x)^(r - 1))}}}
From this, we can see that: {{{matrix(1,4, (- 3x)^(r - 1), "=", highlight("?"), x^2)}}} 
Therefore, r – 1 MUST = 2 
r, or term number with the term 90x<sup>2</sup> = 2 + 1 = 3


Since r = 3, then {{{matrix(1,3, " "[n]C[r - 1] * (- 3x)^(r - 1), "=", 90x^2)}}} becomes:
                  {{{matrix(1,3, ""[n]C[3 - 1] * (- 3x)^(3 - 1), "=", 90x^2)}}} 
                  {{{matrix(1,3, ""[n]C[2] * (- 3x)^2, "=", 90x^2)}}} 
                  {{{matrix(1,3, ""[n]C[2] * 9x^2, "=", 90x^2)}}} 
                  {{{matrix(1,5, ""[n]C[2], "=", 90x^2/9x^2, "=", 10)}}}


{{{matrix(1,3, ""[n]C[r], "=", ""[n]P[r]/r!)}}} 
{{{matrix(1,5, ""[n]C[2], "=", ""[n]P[2]/2!, "=", 10)}}}
{{{matrix(1,3, ""[n]P[2], "=", 20)}}} ------- Cross-multiplying
n(n – 1) = 20
Observing the above equation, it's clear that: {{{highlight_green(matrix(1,4, n, MUST, "=", 5))}}}, as 5(5 – 1), or 5(4) = 20.
You could've also distributed the above equation, and solve the formed quadratic by FACTORING to get the same result: n = 5.</pre>