Question

How do you expand ${{\left( d+5 \right)}^{7}}$ using Pascal’s Triangle?

Answer 1

To solve this, we first need to write the expansion of the given expression ${{\left( d+5 \right)}^{7}}.$ We need to write the first term in decreasing powers whereas the second term in increasing powers. The coefficients for all the terms are determined by the values in the Pascal’s Triangle and we multiply this with every term and write the equation.

Complete step by step solution:
We solve this question by using the concept of binomial expansion and the Pascal’s Triangle. Pascal’s Triangle is nothing but a triangle whose starting value is one and the values of the subsequent rows are determined by the sum of the two terms above it. For example, looking at row four of the Pascal’s Triangle given below, the element 3 is obtained by adding the above two terms which are 1 and 2.

Now for the general binomial expansion for an expression given by ${{\left( x+a \right)}^{n}},$ we can obtain it by writing the equation with decreasing powers of x from n to 0 and increasing powers of a from 0 to n. They also contain coefficients whose values are obtained from the corresponding rows of the Pascal’s Triangle.
For the given equation, we have the power n given as 7. We will have n+1 terms in the expansion, 8 in this case, and have 8 coefficients. The terms are given by decreasing powers of d from 7 to 0 and increasing powers of a from 0 to 7. Therefore, from the given image of the Pascal’s Triangle, we select the last row which has values for the 8 coefficients.
The equation for ${{\left( d+5 \right)}^{7}}$ is expanded as follows, by multiplying the coefficients by the terms.
$\Rightarrow {{\left( d+5 \right)}^{7}}=1.{{d}^{7}}{{.5}^{0}}+7.{{d}^{6}}{{.5}^{1}}+21.{{d}^{5}}{{.5}^{2}}+35.{{d}^{4}}{{.5}^{3}}+35.{{d}^{3}}{{.5}^{4}}+21.{{d}^{2}}{{.5}^{5}}+7.{{d}^{1}}{{.5}^{6}}+1.{{d}^{0}}{{.5}^{7}}$
Multiplying the coefficients with the constant term values and simplifying,
$\Rightarrow {{\left( d+5 \right)}^{7}}={{d}^{7}}+35{{d}^{6}}+525{{d}^{5}}+4375{{d}^{4}}+21875{{d}^{3}}+65625{{d}^{2}}+109375{{d}^{1}}+78125$

Note: While expanding the expression, we can take increasing powers of d and decreasing powers of 5 too. It is the same in either case since the values of coefficients are symmetric with respect to the triangle in the Pascal’s Triangle. Therefore, while expanding the expression, the order of powers increasing or decreasing does not matter.