 # Missing average values

You are given a sequence: 100, X1, X2, 115. We have to find the missing value Xi. It is given that Xi will be average of prev and next value & will be an integer. Here X1 will be 105 and X2 will be 110.
There could be more values missing.

If you think this question is from an ongoing contest please post the link of the contest and I will delete this post. ( I got it from my classmate ).

\dfrac{100+X_2}{2} = X_1

\implies 100 + X_2 = 2X_1

\dfrac{X_1+115}{2} = X_2

\implies X_1 + 115 = 2X_2

100 + X_2 = 4X_2 - 230

330 = 3X_2

X_2 = 110

X_1 = 2(110)-115

X_1=105

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how would you generalise it ?

\dfrac{A+X_2}{2} = X_1

\implies A + X_2 = 2X_1

\dfrac{X_1+B}{2} = X_2

\implies X_1 + B = 2X_2

A + X_2 = 4X_2 - 2B

A+2B = 3X_2

X_2 = \dfrac{A+2B}{3}

X_1 = 2X_2-B

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generalise according to this : A, X1, X2, X3, X4, … , XN , B

Think of it as an arithmetic progression.
We have A, \ X_1, \ X_2, \ \dots,\ X_n, \ B.
There are n+2 terms.
It is easy to see that the common difference d is \dfrac{B-A}{n+1}.
From this, we conclude that X_i = A + id.

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Thank you so much.

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