Question

Seven numbers a_1, a_2, a_3, a_4, a_5, a_6 and a_7 form an arithmetic sequence, where a_1 < a_2 < a_3 < a_4 < a_5 < a_6 < a_7. It is given that the mean of the seven numbers is 0.

Find a_4.

If the number a_4 is deleted, is there any change in the variance of the numbers? Explain your answer.

Collected in the board: Arithmetic sequence

Steven Zheng posted 2 months ago


Answer

Let d be the common difference of the sequence

a_1=a_4-3d

a_2 = a_4-2d

a_3 = a_4-d

a_5=a_4+d

a_6=a_4+2d

a_7 = a_4+3d

a_1+a_2+\dots+a_7

=7a_4=0

Therefore,

a_4 = 0

If a_4 is deleted, previous a_5 becomes the 4th term, previous a_6 becomes the 5th term, previous a_7 becomes 6th term.

a_1=a_4-3d

a_2 = a_4-2d

a_3 = a_4-d

a_4=a_4

a_5=a_4+d

a_6 = a_4+2d

a_1+a_2+\dots+a_6

=6a_4-3d=0

Therefore,

a_4=\dfrac{1}{2}d

which means the numbers vary with the common difference.


Steven Zheng posted 1 month ago

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