Let a 0 , a 1 , … , a 23 be real numbers such that ( 1 + 2 5 x ) 23 = i = 0 23 a i x i for every real number x . Let a r be the largest among the numbers a j for 0 j 23 . Then the value of r is ________. [2025]

Question

Let $a_{0}, a_{1}, \dots, a_{23}$ be real numbers such that ${(1 + \frac{2}{5} x)}^{23} = \sum_{i = 0}^{23} a_{i} x^{i}$ for every real number $x$ . Let $a_{r}$ be the largest among the numbers $a_{j}$ for $0 \leq j \leq 23$ . Then the value of $r$ is ________. [2025]

Smart Achievers · Accepted Answer

(6)

$Put x = 1$

${(1 + \frac{2}{5})}^{23} = a_{0} + a_{1} + a_{2} + \dots + a_{23}$

$For numerically greatest term$

$\frac{n + 1}{1 + | \frac{a}{b} |} = \frac{23 + 1}{1 + \frac{5}{2}} = \frac{48}{7}$

$\Rightarrow [\frac{48}{7}] = 6$ (where $[.]$ denotes greatest integer function)

So, $a_{7}$ is numerically greatest term

Hence, $r = 6$

Solution

Q.
Let $a_{0}, a_{1}, \dots, a_{23}$ be real numbers such that ${(1 + \frac{2}{5} x)}^{23} = \sum_{i = 0}^{23} a_{i} x^{i}$ for every real number $x$ . Let $a_{r}$ be the largest among the numbers $a_{j}$ for $0 \leq j \leq 23$ . Then the value of $r$ is ________. [2025]

Want Us to Email you About Special Offers & Updates?

Solution

Q. Let a0,a1,…,a23 be real numbers such that (1+25x)23=∑i=023aixi for every real number x. Let ar be the largest among the numbers aj for 0≤j≤23. Then the value of r is ________. [2025]

Ans. (6) Put x=1 (1+25)23=a0+a1+a2+⋯+a23 For numerically greatest term n+11+|ab|=23+11+52=487 ⇒[487]=6 (where [.] denotes greatest integer function) So, a7 is numerically greatest term Hence, r=6

Want Us to Email you About Special Offers & Updates?

Q.
Let $a_{0}, a_{1}, \dots, a_{23}$ be real numbers such that ${(1 + \frac{2}{5} x)}^{23} = \sum_{i = 0}^{23} a_{i} x^{i}$ for every real number $x$ . Let $a_{r}$ be the largest among the numbers $a_{j}$ for $0 \leq j \leq 23$ . Then the value of $r$ is ________. [2025]