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Thermal Expansion

For r=0,1,2,....

Question

For r=0,1,2,........10, let $A_{r}, B_{r}$ and $C_{r}$ denote respectively the coefficient of $x^{r}$ in the expansions of $(1 + x)^{10}, (1 + x)^{20}$ and $(1 + x)^{30}$ . Then $10 \sum r = 1 A_{r} (B_{10} B_{r} - C_{10} A_{r})$ is

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Solution

$10 \sum r = 1 A_{r} (B_{10} B_{r} - C_{10} A_{r})$
$= 10 \sum r = 1 (B_{10} A_{r} B_{r} - C_{10} A_{r}^{2}) . . . . (1)$
Now, $10 \sum r = 1 A_{r} B_{r} =$ co-efficient of $x^{20}$ in $((1 + x)^{10} (x + 1)^{20}) - 1$
$= C_{20} - 1$
$= C_{10} - 1...... (2)$
$10 \sum r = 1 (A_{r})^{2} =$ co-efficient of $x^{10}$ in $((1 + x)^{10} (x + 1)^{10}) - 1$
$= B_{10} - 1..... (3)$
Substituting both $(2)$ and $(3)$ in $(1)$
$= B_{10} (C_{10} - 1) - C_{10} (B_{10} - 1)$
$= C_{10} - B_{10}$

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r = 0, 1, 2, 3, . . ., 10

A_{r}, B_{r}, C_{r}

denote respectively the coefficient of

x^{r}

in the expansions of

(1 + x)^{10}, (1 + x)^{20}

(1 + x)^{30}

\sum_{r = 1}^{10} A_{r} (B_{10} B_{r} - C_{10} A_{r})

r = 0, 1, . . ., 10

A_{r}, B_{r}

C_{r}

denote, respectively, the coefficient of

x^{r}

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(1 + x)^{30}

10 \sum r = 1 A_{r} (B_{10} B_{r} - C_{10} A_{r})

r = 0, 1, \dots

A_{r}, B_{r}

C_{r}

denote, respectively, the coefficient of

x^{r}

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(1 + x)^{10}, (1 + x)^{20}

(1 + x)^{30}

10 \sum r = 1 A_{r} (B_{10} B_{r} - C_{10} A_{r})

r = 0, 1, . . ., 10

A_{r}, B_{r},

C_{r}

denote, respctively, the coefficient of

x^{r}

in the expansions of

(1 + x)^{10}, (1 + x)^{20}

(1 + x)^{30}

\sum_{r = 1}^{10} A_{r} (B_{10} B_{r} - C_{10} A_{r})

r = 0, 1, 2,, . . . .10

A_{r}, B_{r}

C_{r}

denote respectively the coefficient of

x^{r}

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\sum_{r = 1}^{10} A_{r} (B_{10} B_{r} - C_{10} A_{r})

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