Discussion

Question:

If A(a₁, a₂, …a_n) = Average of (a₁, a₂, …, a_n) and A’_n = A(n, n²), then what is the value of A(A’₁, A’₂, …, A’_n), where n = 100?

Explanation:

A’_n = A(n, n²) = (n + n²)/2,

A(A’₁, A’₂, …, A’₁₀₀) = A $(\frac{1 + 1^{2}}{2}, \frac{2 + 2^{2}}{2}, \frac{3 + 3^{2}}{2}, . . ., \frac{100 + 100^{2}}{2})$ .

= $\frac{(\frac{1 + 1^{2}}{2}, \frac{2 + 2^{2}}{2}, \frac{3 + 3^{2}}{2}, . . ., \frac{100 + 100^{2}}{2})}{100}$

= $\frac{(1 + 2 + . . . + 100) + (1^{2} + 2^{2} + . . . + 100^{2})}{200}$

= $\frac{\frac{100 \times 101}{2} + \frac{100 \times 101 \times 201}{6}}{200}$

= $\frac{101}{4} (1 + \frac{201}{3})$

= $\frac{101}{4} \times 68$

= 1717

Hence, 1717.

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