If the sum of the coefficients of all the positive powers of x, in the binomial expansion of (xⁿ + 2

Q: Wait, why do we stop at $r=4$? How do we know there aren't more terms with positive powers later?

That is a brilliant observation! We stop at $r=4$ because the problem explicitly tells us the sum of coefficients of *all* positive powers is 939. If there were another term with a positive power (say, at $r=5$), its coefficient would have been added to the sum, making it greater than 939. Since we hit exactly 939 at $r=4$, we know that $r=4$ is the last term with a positive power.

Q: Why is the condition for $P(5)$ less than or equal to zero, and not just strictly less than zero?

Great catch! If $P(5)$ were exactly zero, the power of $x$ would be $x^0$, which is 1. This is a constant term, not a 'positive power' of $x$. Since the problem asks for positive powers, $x^0$ is excluded. Therefore, $P(5) \le 0$ is the correct boundary condition to ensure the term at $r=5$ is not included in our sum.

Q: Why can't we just use the Binomial Theorem expansion formula directly to find the sum?

You could, but that would give you the sum of *all* coefficients in the expansion. Here, we have a filter: we only want coefficients where the power of $x$ is positive. This makes it a conditional sum, not a total sum. We must inspect the terms one by one to see which ones pass the 'positive power' test.

Question

If the sum of the coefficients of all the positive powers of $x$, in the binomial expansion of $\left( x^n + \frac{2}{x^5} \right)^7$ is 939, then the sum of all the possible integral values of $n$ is :

Sigma Percentile · Accepted Answer

### Analyzing the General Term

To begin, we define the general term $T_{r+1}$ of the binomial expansion $(x^n + \frac{2}{x^5})^7$. Using the Binomial Theorem, the expression is given by:

$$T_{r+1} = {}^7C_r \cdot (x^n)^{7-r} \cdot \left( \frac{2}{x^5} \right)^r$$

By simplifying this expression, we separate the numerical coefficients from the variable powers:

$$T_{r+1} = ({}^7C_r \cdot 2^r) \cdot x^{n(7-r) - 5r}$$

The exponent of $x$, which we denote as $P(r)$, is defined by the function:

$$P(r) = n(7-r) - 5r$$

### The Coefficient Game

We are interested in the sum of coefficients of the positive powers of $x$. Let $C_r = {}^7C_r \cdot 2^r$ represent the coefficient of the $(r+1)$-th term. We calculate these values for $r = 0, 1, 2, 3, 4$:

For $r=0$: $C_0 = {}^7C_0 \cdot 2^0 = 1 \cdot 1 = 1$

For $r=1$: $C_1 = {}^7C_1 \cdot 2^1 = 7 \cdot 2 = 14$

For $r=2$: $C_2 = {}^7C_2 \cdot 2^2 = 21 \cdot 4 = 84$

For $r=3$: $C_3 = {}^7C_3 \cdot 2^3 = 35 \cdot 8 = 280$

For $r=4$: $C_4 = {}^7C_4 \cdot 2^4 = 35 \cdot 16 = 560$

Summing these coefficients, we find:

$$1 + 14 + 84 + 280 + 560 = 939$$

### Establishing Boundary Conditions

Since the sum of coefficients for terms with positive powers is exactly 939, the terms from $r=0$ to $r=4$ must correspond to positive powers of $x$. This implies two critical constraints on $n$.

First, the term at $r=4$ must have a positive power ($P(4) > 0$):

$$n(7-4) - 5(4) > 0 \Rightarrow 3n - 20 > 0 \Rightarrow n > \frac{20}{3} \approx 6.66$$

Second, the term at $r=5$ must not have a positive power ($P(5) \le 0$), otherwise, its coefficient would have been included in the sum:

$$n(7-5) - 5(5) \le 0 \Rightarrow 2n - 25 \le 0 \Rightarrow n \le 12.5$$

### Final Calculation

Combining these inequalities, we obtain the range for $n$:

$$6.66 < n \le 12.5$$

The integers $n$ that satisfy this condition are $7, 8, 9, 10, 11,$ and $12$. The sum of these values is:

$$7 + 8 + 9 + 10 + 11 + 12 = \mathbf{57}$$

Animated Solution for Mathematics - Binomial Theorem: If the sum of the coefficients of all the positive powers of $x$ , in the binomial expansion of $(x^{n} + \frac{2}{x ^{5}})^{7}$ is 939, then the sum of all the possible integral values of $n$ is :

Enter Numerical Value:

Visualized Solution

Define the General Term $T_{r + 1}$

Simplify the Power of $x$

Identify the Coefficients $C_{r}$

Calculate Coefficients $C_{0}$ to $C_{2}$

Calculate Coefficients $C_{3}$ and $C_{4}$

Establish the Boundary Condition

Solve the First Inequality

Solve the Second Inequality

Find Integral Values of $n$

Final Sum Calculation

The Sigma Insight: General Term and Middle Term

Analyzing the General Term

The Coefficient Game

Establishing Boundary Conditions

Final Calculation

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Enter Numerical Value:

Visualized Solution

Define the General Term .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Tr+1​

Simplify the Power of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x

Identify the Coefficients .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Cr​

Calculate Coefficients .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C0​ to .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C2​

Calculate Coefficients .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C3​ and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C4​