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Question:

The average of a non-decreasing sequence of N number a₁, a₂, ..., a_N is 300. If a1 is replaced by 6a₁, the new average becomes 400. Then, the number of possible values of a₁ is

Explanation:

Given, $\frac{a_{1} + a_{2} + . . . + a_{N}}{N}$ = 300

⇒ a₁ + a₂ + ... + a_N = 300N ...(1)

Also, $\frac{6 a_{1} + a_{2} + . . . + a_{N}}{N}$ = 400

⇒ 6a₁ + a₂ + ... + a_N = 400N ...(2)

(2) - (1)

⇒ 5a₁ = 100N

⇒ a₁ = 20N

Since, a₁ is the least of the given numbers it cannot be more than the average, hence a₁ ≤ 300.

⇒ N ≤ 15

If N = 1, a₁ = 20 and average cannot be equal to 300.

Hence, N can take all values from 2 till 15, i.e., 14 values.

Hence, 14.

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Doubts

Chaitanya said (2023-10-26 11:40:24)

If a1=20 why can't the average be300

Reply from Admin:

If N = 1, a₁ = 20.

Average of just one term (20) cannot be 300.

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