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Arithmetico-Geometric Sequence

Let a, (a + d)r, (a + 2d)r2, (a + 3d)r3, …, be an arithmetic- geometric sequence.
 
Then, a + (a + d)r + (a + 2d)r2 + (a + 3d)r3 + is an arithmetico-geometric series.
 
Sum of n terms of an arithmetico-geometric sequence
62965.png(1)
If the students find it difficult to apply this formula, then it is advised to find the sum by the following mechanism:
 
Write the series as sum S = a + (a + d)r + (a + 2d)r2 + (a + 3d)r3 +
 
Now write the series after multiplying by common ratio r, as rS = ar + (a + d)r2 + (a + 2d)r3 + (a + 3d)r4 +
 
Now subtract the rS from S and find the sum of resulting GP.
 
Sum of an infinite arithmetico-geometric sequence
 
If |r| < 1, then rn, rn–1 0 as n ∞ and it can also be shown that n rn 0 as n ∞. So, from (1), we obtain that Sn 62959.png, as n ∞.

Method of difference

Consider the sequence S : a1, a2, a3, a4, ….
 
If difference series a2a1, a3a2, a4a3, … is AP,
then general term of S is an = An2 + Bn + C, where A, B, C are constant.
We can find values of A, B, C by putting n = 1, 2, 3.
 
If difference series a2a1, a3a2, a4a3, … is GP with common ratio r, then general term of S is an = Arn + Bn + C, where A, B, C are constants.
We can find values of A, B, C by putting n = 1, 2, 3.

 

Notes: For any series if Sn, sum of n terms, is given then we can find the nth term from tn = Sn – Sn – 1.





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