Question

If ( x - 1 )¹⁰⁰ = a₀ + a₁x + a₂x² .... + a₁₀₀x¹⁰⁰

Then find :-
i ) The value of a₀
ii ) a₁ + a₂ + a₃ + a₄ .... + a₁₀₀

Answer 1

$\large\underline{\sf{Given- }}$

• ( x - 1 )¹⁰⁰ = a₀ + a₁x + a₂x² .... + a₁₀₀x¹⁰⁰

$\large\underline{\sf{To\:Find - }}$

• (i) a₀

• (ii) a₁ + a₂ + a₃ + a₄ .... + a₁₀₀

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\: {(x - 1)}^{100} = a_0 + a_1x + a_2 {x}^{2} + - - - + a_{100} {x}^{100}$

Now, to get the value of a₀.

Substitute x = 0 in the given equation, we get

$\rm :\longmapsto\: {(0 - 1)}^{100} = a_0 + a_1(0) + a_2 {(0)}^{2} + - - - + a_{100} {(0)}^{100}$

$\rm :\longmapsto\: {(- 1)}^{100} = a_0$

We know,

$\underbrace{ \boxed{ \bf \: {( - 1)}^{even \: natural \: number} = 1}}$

So,

$\bf\implies \:a_0 = 1$

Now,

To find the value of a₁ + a₂ + a₃ + a₄ .... + a₁₀₀

We have given that,

$\rm :\longmapsto\: {(x - 1)}^{100} = a_0 + a_1x + a_2 {x}^{2} + - - - + a_{100} {x}^{100}$

On substituting x = 1, we get

$\rm :\longmapsto\: {(1 - 1)}^{100} = a_0 + a_1(1) + a_2 {(1)}^{2} + - - - + a_{100} {(1)}^{100}$

$\rm :\longmapsto\: {(0)}^{100} = a_0 + a_1 + a_2 + - - - + a_{100}$

$\rm :\longmapsto\: a_0 + a_1 + a_2 + - - - + a_{100} = 0$

As, we proved above

$\rm :\longmapsto\:a_0 = 1$

So, on substituting the value of a₀, we get

$\rm :\longmapsto\: 1 + a_1 + a_2 + - - - + a_{100} = 0$

$\rm :\longmapsto\: a_1 + a_2 + - - - + a_{100} = 0 - 1$

$\rm :\longmapsto\: a_1 + a_2 + - - - + a_{100} = - 1$

Hence,

$\rm :\longmapsto\: If \: {(x - 1)}^{100} = a_0 + a_1x + a_2 {x}^{2} + - - + a_{100} {x}^{100}$

then

$\rm :\longmapsto\:a_0 = 1$

and

$\rm :\longmapsto\: a_1 + a_2 + - - - + a_{100} = - 1$

Let solve one more problem of same type!!

Question

$\rm :\longmapsto\: If \: {(x - 1)}^{100} = a_0 + a_1x + a_2 {x}^{2} +a_3 {x}^{3} - + a_{100} {x}^{100}$

find the value of

$\rm :\longmapsto\:a_0 - a_1 + a_2 - a_3 + - - - + a_{100}$

Solution :-

Given that

$\rm :\longmapsto\: {(x - 1)}^{100} = a_0 + a_1x + a_2 {x}^{2} +a_3 {x}^{3} + - - + a_{100} {x}^{100}$

To find the value,

Put x = - 1 in the given equation, we get

$\rm :\longmapsto\: {( - 1 - 1)}^{100} = a_0 + a_1( - 1) + a_2 {( - 1)}^{2} +a_3 {( - 1)}^{3} + - - + a_{100} {( - 1)}^{100}$

$\rm :\longmapsto\: {( - 2)}^{100} = a_0 - a_1+ a_2 - a_3 + - - - + a_{100}$

$\rm :\longmapsto\: a_0 - a_1+ a_2 - a_3 + - - - + a_{100} = {2}^{100}$