Arithmeticogeometric Progression Solved Examples

As the name suggests, an arithmeticogeometric progression is obtained when the corresponding terms of a geometric and arithmetic progression are multiplied by each other.

Mathematically, it can be expressed as

x_n = b_nh_n, where

x_n is the nth term of the arithmeticogeometric progression

b_n is the nth term of the arithmetic progression

h_n is the nth term of the geometric progression

Illustration:

Find the sum of series 1. 2 + 2. 2² + 3. 2³ +…+ 100. 2¹⁰⁰.

Solution: Let us denote the given series by S.

Let S = 1.2 + 2.2² + 3.2³ +…+ 100.2¹⁰⁰                           …… (1)

⇒ 2S = 1.2² + 2.2³ +…+ 99.2¹⁰⁰ + 100.2¹⁰¹                      …… (2)

⇒ –S = 1.2 + 1.2² + 1.2³ +…+ 1.2¹⁰⁰ – 100.2¹⁰¹

⇒ –S = 1.2 (2⁹⁹–1/(2–1)) – 100.2¹⁰¹

⇒ S = –2¹⁰⁰ + 2 + 100.2¹⁰¹ = 199.2¹⁰⁰ + 2. 

Illustration:

Let r = 1/2, consider nrⁿ for increasing value of n and then find its value as n tends to infinity.

Solution: We first consider n (1/2)ⁿ for increasing values of n.

       n = 1 we get 1. (1/2)¹ = 1/2 = 0.5

       n = 2 gives 2 × (1/2)² = 1/2 = 0.5

       n = 3 gives 3 × (1/2)³ = 0.375

       n = 10 : 10 (1/2)¹⁰ = 0.00976, and so on

Thus we observe that as n → ∞
        n rⁿ → 0 for |r| < 1.

Illustration:

Evaluate 1 + 4/5 + 7/5² + 10/5³ +…… to infinite terms.

Solution: We denote the given sequence by S.

        Let S = 1 + 4/5 + 7/5² + 10/5³ + ………

        1/5 S = 1/5 + 4/5² + 7/5³ ………

        Subtracting

        (1–1/5) S = 1 + 3/5 + 3/5² + 3/5³ + ………

        4/5 S = 1/ (1–3/5)                             (As it is infinite G.P.)

        ⇒ S =25/8 

Refer the following video for more on arithmeticogeometric progression

Let t₁, t₂, t₃, ……, t_m–1, t_m, t_m+1, be a sequence so that

 (i) t_m+1/t_m = t_m/t_m–1 ……… constant                             (r)

       then t_p = (t₁)r^p–1
 (ii) t_m+1/t_m = t_m/t_m–1 = constant                                  (r)

       then t_p = constant 1 + (constant 2) × r^p–1

 (iii) If the difference of difference of terms are in G.P. then

        t_p = a + bp + cr^p–1, where r is the common ratio. 

Illustration:

7. 14. 33. 88. 251. 738 …………

Solution: We can represent the given sequence in the form of a diagram as shown   

Remark: First we write down all the numbers in a straight line. Then we find out the difference between every pair of two terms. This is denoted in the second line. Again we find the difference between every pair of consecutive terms in the second line. Now we divide every successive term by the preceding term and see that the ratio in each case is 3.  

       324/108 = 108/36 = 36/12 = 3

Hence, using the formula listed above

∴      t_p = a + bp + c 3^p–1

        p=1 so t₁ = 7 = a + b + c

        p=2 so t₂ = 14 = a + 2b + 3c

        p=3 so t₃ = 33 = a + 3b + 9c

Solving, we get a = 3, b = 1, c = 3

⇒     t_p = 3 + p + 3. (3^p–1)
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