If [tex](x+y)^4 = K_1 x^4 + K_2 x^3 y + K_3 x^2 y^2 + K_4 x y^3 + K_5 y^4[/tex], find the value of:

(i) [tex]\sum_{i=1}^5 K_i[/tex]

(ii) [tex]2 \sum_{i=2}^4 K_i[/tex]

(iii) [tex]3 \sum_{i=1}^4 K_i - 2 \sum_{i=3}^5 K_i[/tex]



Answer :

To address the given problem, we start by understanding the expansion of [tex]\((x + y)^4\)[/tex]:

[tex]\[ (x + y)^4 = K_1 x^4 + K_2 x^3 y + K_3 x^2 y^2 + K_4 xy^3 + K_5 y^4 \][/tex]

From the binomial theorem, we know the coefficients are the binomial coefficients:
[tex]\[ (x + y)^4 = \sum_{i=0}^{4} \binom{4}{i} x^{4-i} y^i \][/tex]

Therefore,
[tex]\[ K_1 = \binom{4}{0} = 1, \quad K_2 = \binom{4}{1} = 4, \quad K_3 = \binom{4}{2} = 6, \quad K_4 = \binom{4}{3} = 4, \quad K_5 = \binom{4}{4} = 1 \][/tex]

Now, let's define the sums required for the question:
1. [tex]\(\mu \sum_{i=1}^5 K_i\)[/tex]
2. [tex]\(2 \sum_{i=2}^4 K_i\)[/tex]
3. [tex]\(3 \sum_{i=1}^4 K_i - 2 \sum_{i=3}^5 K_i\)[/tex]

### Step-by-step solution:

1. Given coefficients:

[tex]\[ K_1 = 1, \quad K_2 = 4, \quad K_3 = 6, \quad K_4 = 4, \quad K_5 = 1 \][/tex]

2. Calculating each required expression:

i. [tex]\(\mu \sum_{i=1}^5 K_i\)[/tex]

[tex]\[ \sum_{i=1}^5 K_i = K_1 + K_2 + K_3 + K_4 + K_5 = 1 + 4 + 6 + 4 + 1 = 16 \][/tex]
Assuming [tex]\(\mu = 1\)[/tex]

[tex]\[ \mu \sum_{i=1}^5 K_i = 1 \cdot 16 = 16 \][/tex]

ii. [tex]\(2 \sum_{i=2}^4 K_i\)[/tex]

[tex]\[ \sum_{i=2}^4 K_i = K_2 + K_3 + K_4 = 4 + 6 + 4 = 14 \][/tex]

[tex]\[ 2 \sum_{i=2}^4 K_i = 2 \cdot 14 = 28 \][/tex]

iii. [tex]\(3 \sum_{i=1}^4 K_i - 2 \sum_{i=3}^5 K_i\)[/tex]

[tex]\[ \sum_{i=1}^4 K_i = K_1 + K_2 + K_3 + K_4 = 1 + 4 + 6 + 4 = 15 \][/tex]

[tex]\[ \sum_{i=3}^5 K_i = K_3 + K_4 + K_5 = 6 + 4 + 1 = 11 \][/tex]

[tex]\[ 3 \sum_{i=1}^4 K_i - 2 \sum_{i=3}^5 K_i = 3 \cdot 15 - 2 \cdot 11 = 45 - 22 = 23 \][/tex]

### Final Results:

1. [tex]\(\mu \sum_{i=1}^5 K_i = 16\)[/tex]
2. [tex]\(2 \sum_{i=2}^4 K_i = 28\)[/tex]
3. [tex]\(3 \sum_{i=1}^4 K_i - 2 \sum_{i=3}^5 K_i = 23\)[/tex]

Thus, the values are:

(i) [tex]\( \mu \sum_{i=1}^5 K_i = 16 \)[/tex]

(ii) [tex]\( 2 \sum_{i=2}^4 K_i = 28 \)[/tex]

(iii) [tex]\( 3 \sum_{i=1}^4 K_i - 2 \sum_{i=3}^5 K_i = 23 \)[/tex]