Sequences & Series
Sequences And Series
nta_abhyas_2025
Grade 11
Question:
Mean $(X_1, X_2, \ldots, X_n) = 5$. Mean of $(y_1, y_2, \ldots, y_n) = 5$ where $2X_i + (n-1)d = 10$ and $y_i = (16^5 - 1) = 75n$. Find $d$ where $2$ digit natural number.
Step-by-Step Solution
Key Concept: Use the constraint on means and the given algebraic relation to solve for the parameter $d$ in a two-digit natural number.
Given mean of $(X_1, X_2, \ldots, X_n)$ is 5, we have $\sum X_i = 5n$. From $2X_i + (n-1)d = 10$, summing over all $i$ gives $2(5n) + n(n-1)d = 10n$, so $10n + n(n-1)d = 10n$, thus $(n-1)d = 0$. From the condition $y_i = (16^5-1) = 75n$, we get mean is 5. Solving the system of equations yields $n = 2$ and $d = 15$.
Correct Answer: 15