Arithmetico-Geometric Series

Arithmetico – Geometric Series

Suppose a₁, a_2, . . ., an are in AP and b₁, b₂, . . . bn are in GP then an expression of the type a₁ b₁, a₂ b_2. . .an bn is said to be an Arithmetico Geometric Series and of the form

S_n = ab + (a+d) br + (a+2d) br₂ + . . . + (a+n-1) d . × br_n-1 …(1)

We multiply each term by the common ratio of GP.

r × SN = abr + (a+d) br₂ + . . . + {a + (n-1) d} br_n …(2)

Subtracting (1) and (2) and simplifying we get

If |r| < 1 then

Then the above mentioned sum reduces to .