Question

Choose seven consecutive terms in any arithmetic sequence and check that the arithmetic average (the sum of the seven terms divided by 7) equals the middle term. Prove that the result is true for every arithmetic sequence. give me a paragraph to explain.

Answer 1

Let 1,4,7,10,13,16,19 is an arithmetic sequence with common difference 3.

Arithmetic average = (1+4+7+10+13+16+19)/7

Arithmetic average = 70/7

Arithmetic average = 10 which is middle term.

To prove result in general :

X1,X2,X3,X4,X5,X6,X7 is an arithmetic sequence with common difference "a".

Then, X2 = X1+a

X3 = X1+2a

X4 = X1+3a

X5 = X1+4a

X6 = X1+5a

X7 = X1+6a

Arithmetic average = (X1+X2+X3+X4+X5+X6+X7)/7

Arithmetic average = (X1+X1+a+X1+2a+X1+3a+X1+4a+X1+5a+X1+6a)/7

Arithmetic average = (7X1+21a)/7

Arithmetic average = 7(X1+3a)/7

Arithmetic average = X1+3a

therefore arithmetic average is equal to middle term.