**How to Find the Sum of Special Type of Geometric Series ?**

Here we are going to see how to find geometric series with special types.

**Question 1 :**

If the first term of an infinite G.P. is 8 and its sum to infinity is 32/3 then find the common ratio.

**Solution :**

Since the given sequence is infinite series, the sum of the series = a/(1 - r)

a/(1-r) = 32/3

a = 8

8/(1-r) = 32/3

8(3) = 32(1-r)

24 / 32 = 1 - r

3/4 = 1 - r

r = 1 - (3/4) ==> 1/4

**Question 2 :**

Find the sum to n terms of the series

(i) 0.4 + 0.44 + 0.444 +........ to n terms

**Solution :**

** 0.4 + 0.44 + 0.444 +........ to n terms**

** = 4(0.1 + 0.11 + 0.111 + ...... to n terms)**

= 4 ⋅ (9/9) (0.1 + 0.11 + 0.111 + ...... to n terms)

= (4/9)[0.9 + 0.99 + 0.999 +............n terms]

= (4/9)[(1 - 0.1) + (1 - 0.1^{2}) + (1 - 0.1^{3}) + .......n terms]

= (4/9)[(1+1+1........n terms) - [0.1+0.1^{2}+0.1^{3} + .......n terms]

= (4/9)[n - 0.1 [1 - (0.1)^{n}/(1-0.1)]

= (4/9)[n - 0.1 [1 - (0.1)^{n}/0.9]

= (4/9)[n - [(1 - (0.1)^{n})/0.9]

(ii) 3 + 33 + 333 + ........... to n terms

**Solution :**

3 + 33 + 333 + ........... to n terms

= 3 [1 + 11 + 111 + ...........+ n terms]

= 3 ⋅ (9/9) (1 + 11 + 111 + ...... to n terms)

= (3/9)[9 + 99 + 999 +............n terms]

= (1/3)[(10-1) + (100 - 1) + (1000 - 1) +............n terms]

=(1/3)[(10+100+1000 + .......n terms)-(1+1+1+.........n terms)]

a = 10, r = 100/10 = 10, a = 1 and r = 1

= (1/3)[10(10^{n} - 1)/(10-1))-n]

= (1/3)[(10/9)(10^{n} - 1) - n]

= [(10/27)(10^{n} - 1) - (n/3)]

**Question 3 :**

Find the sum of the Geometric series

3 + 6 + 12 +............+ 1536

**Solution :**

t_{n} = 1536

a = 3, r = 6/3 = 2

ar^{n-1} = 1536

3(2)^{n-1} = 1536

2^{n-1} = 1536/3

2^{n-1} = 512

2^{n-1} = 2^{9}

n - 1 = 9

n = 10

Now, we have to find the sum of 10 terms.

Sn = a(r^{n} - 1)/(r - 1)

S_{10 } = 3(2^{10} - 1)/(2 - 1)

= 3(1024 - 1)

= 3(1023)

S_{10 } = 3069

After having gone through the stuff given above, we hope that the students would have understood, "How to Find the Sum of Special Type of Geometric Series".

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