Tag Archives: Geometric progression

Triangle (a,b,c) – inradius and circumradius — (Part 2)

    Establish the following For any triangle   ,     and where     is the circumradius,     the inradius and     the semiperimeter. Then     are in geometric progression iff           … Continue reading

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To make {(A+n),(B+n),(C+n)} squares, (A,B,C) in geometrical progression

    Find three numbers     in geometrical progression such that each increased by a given number   is a square number.                                   … Continue reading

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When (X,Y,Z) in geometrical progression {(X-Y), (X-Z), (Y-Z)} squares

    To find three rational numbers in geometrical progression, the difference of any two of which is a square number.     Here’s one possible solution: The integers, form a Pythagorean triple.   and the following integers, are in … Continue reading

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Sum of a geometric series : 1 + r + r^2 + … + r^n = A^2

  The general form of a geometric sequence is   is the first term, and   is the factor between the terms (called the “common ratio”) Find a geometric series of 3 or more positive integers, starting with 1, such … Continue reading

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Sets of triangular numbers in geometric progression

    In a geometric progression, any three consecutive terms   a,   b   and   c will satisfy the following equation:     common ratio = 6,     ,      ,      ,      common ratio = 35/6 … Continue reading

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Sets of 3 triangular numbers in geometric progression

    In a geometric progression, any three consecutive terms   a,   b   and   c will satisfy the following equation:     common ratio = 6,     ,      ,      ,      common ratio = 35/6 … Continue reading

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Geometric progression (1+k+k^2+k^3+k^4)(1-k+k^2-k^3+k^4)

  &nbsp In general we write a geometric sequence like this: where:   is the first term, and   is the factor between the terms (called the “common ratio”) sum of the first few terms of sequence: Using the identity: … Continue reading

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(a,b) : first and last term of arithmetic and geometric progressions

    If      is an arithmetic progression (AP)    is a geometric progression (GP) Show that                          

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Geometric progression – some interesting results

  The general form of a geometric sequence is ,   ,   ,   ,   The n-th term :     where   r ≠ 0   is the common ratio and a is a scale factor, equal to … Continue reading

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(a,b,c) in geometric sequence; a+b=c+1

    To find three numbers   a,   b,   and   c   in geometric sequence with   a + b = c + 1     Additional condition:     a < b < c       … Continue reading

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