Arithmetic Sequences

Consider all increasing arithmetic sequences with common difference 1, {t₁,t₂,t₃...}. If the absolute difference of

100

Σt_x

x=1

and

Σt_2x

x=1

is 1100, find the sum of all possible values of

Σt_4x

x=1

View the solution

Arithmetic Sequences Solution

Let t₁ = a
t_n = a + (n-1)
t₂ = a + 1
t₄ = a + 3
t₁₀₀ = a+99

(a+(a+99))

100

= 50(2a+99)
((a+1)+(a+99))

= 25(2a+100)

|50(2a+99)-25(2a+100)| = |50(a+49)|=1100

a + 49 = 22 or -22

a = -27 or -71

((a+3)+(a+99))

= 25(a+51) = 600 or -500

600 - 500 = 100

What is this number?

Expansion

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