If ∑r=1^9 ((r+3)/2^r) · 9Cr = α (3/2)^9 - β, α, β ∈ N, then (α+β)^2 is equal to

The correct answer is Option 4. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q8 – JEE-MAIN 2025

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Home Maths Binomial TheoremQuestion #45

\sum_{r = 1}^{9} (\frac{r + 3}{2^{r}}) \cdot^{9} C_{r} = α {(\frac{3}{2})}^{9} - β

α, β \in N

, then

(α + β)^{2}

is equal to

Clue 1 of 3: Binomial Theorem Properties

The sum involves binomial coefficients and powers, suggesting the use of binomial expansion properties.

🧠 Full Solution Path

Step 1: Decompose the Sum✦ Active

Split the given summation into two parts based on the numerator $(r + 3)$ :

S = \sum_{r = 1}^{9} (\frac{r + 3}{2^{r}})^{9} C_{r} = \sum_{r = 1}^{9} \frac{r}{2^{r}}^{9} C_{r} + 3 \sum_{r = 1}^{9} \frac{1}{2^{r}}^{9} C_{r}

Step 2: Apply Binomial Summation Identities○ Expand

Use the identities $\sum_{r = 1}^{n} r^{n} C_{r} x^{r} = n x (1 + x)^{n - 1}$ and $\sum_{r = 1}^{n}^{n} C_{r} x^{r} = (1 + x)^{n} - 1$ with $n = 9$ and $x = \frac{1}{2}$ .

S_{1} = \sum_{r = 1}^{9} r^{9} C_{r} {(\frac{1}{2})}^{r} = 9 \cdot \frac{1}{2} {(1 + \frac{1}{2})}^{9 - 1} = \frac{9}{2} {(\frac{3}{2})}^{8}

S_{2} = 3 \sum_{r = 1}^{9}^{9} C_{r} {(\frac{1}{2})}^{r} = 3 [{(1 + \frac{1}{2})}^{9} - 1] = 3 [{(\frac{3}{2})}^{9} - 1]

💡 Teacher's Secret Hint

Remember that the sum for $r = 0$ is excluded in the given problem, so adjust the standard binomial expansion accordingly.

Step 3: Combine and Simplify○ Expand

Substitute $S_{1}$ and $S_{2}$ back into the original sum and simplify to match the target form $α {(\frac{3}{2})}^{9} - β$ :

S = S_{1} + S_{2} = \frac{9}{2} {(\frac{3}{2})}^{8} + 3 [{(\frac{3}{2})}^{9} - 1]

Rewrite ${(\frac{3}{2})}^{8} = {(\frac{3}{2})}^{9} \cdot (\frac{2}{3})$ :

S = \frac{9}{2} \cdot {(\frac{3}{2})}^{9} \cdot (\frac{2}{3}) + 3 {(\frac{3}{2})}^{9} - 3

S = 3 {(\frac{3}{2})}^{9} + 3 {(\frac{3}{2})}^{9} - 3

S = 6 {(\frac{3}{2})}^{9} - 3

Comparing this with $α {(\frac{3}{2})}^{9} - β$ , we get $α = 6$ and $β = 3$ . Both are natural numbers. Finally, calculate $(α + β)^{2}$ :

(α + β)^{2} = (6 + 3)^{2} = 9^{2} = 81

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Maths Question 8 – JEE-MAIN 2025

Clue 1 of 3: Binomial Theorem Properties

🧠 Full Solution Path