**Banking Exam Preparation – ADDITION and SUBTRACTION- Part III**

Banking Exam Preparation – ADDITION and SUBTRACTION- Part III

**Continue from Banking Exam Preparation – ADDITION and SUBTRACTION- Part II**

**Geometric Series**

E.g. 2,4,8,16,32

Each term in series is constant multiple of proceeding term. Constant multiplier is called common ratio.

Constant multiplier (cm) may be less than one or greater than one.

Means cm=1, or cm>1 or cm<1

For e.g. cm=2 means cm>1

**If cm ≥ 1 then**

Sum of a geometric series

Formula **a(cm ^{n} – 1)/cm – 1**

Where a = First number of the series

cm = Constant multiplier

n = Number of elements in series

** Means find n ^{th} root of the constant multiplier and less one form this. After that multiply the result by the first number of the series and divide the result by constant multiplier less one.**

**If cm < 1 then**

Sum of a geometric seires

Formula **a(1 – cm ^{n})/1 – cm**

Where a = First number of the series

cm = Constant multiplier

n = Number of elements in series

**Means find n ^{th} root of the constant multiplier and less this from one. After that multiply the result by the first number of the series and divide the result by one less constant multiplier.**

**Banking Exam Preparation – ADDITION and SUBTRACTION- Part III**

**Sum of Infinite geometric series**

Formula – **a/(1-cm)**, when cm < 1

**Banking Exam Preparation – ADDITION and SUBTRACTION- Part III**

**Example:-**

- 3+33+333+3333+33333

**Solve-** 3+33+333+3333+33333

Take 3 as common from series

= 3{1+11+111+1111+11111}

Multiply and divide 9 in series

= (3/9) x9{1+11+111+1111+11111}

= 3/9{9+99+999+9999+99999}

= 3/9{(10-1) + (100-1) + (1000-1) + (10000-1) + (100000-1)}

= 3/9{(10^{1}-1) + (10^{2}-1) + (10^{3}-1) + (10^{4}-1) + (10^{5}-1)}

For total of (-1) is -5

= 3/9{(10 + 10^{2} + 10^{3} + 10^{4} + 10^{5})-5}

Now we get a geometric series 10 + 10^{2} + 10^{3} + 10^{4} + 10^{5} for this series we use formula

= 3/9[{10(10^{5}-1)/10-1}-5]

= 3/9{111110-5}

= 3/9{111105)

= 3×12345

= 37035

Formula (reminder) –

**Common Digit/9(Solution of geometric series of 10 – n)**

**Common digit/9{10/9x(10 ^{n}-1) – n}**

**Shortcut – Solution of geometric series of 10- if constant multiplier is 10 and first number is also 10 then place ‘1’ for each ‘n’ and add ‘0’ in place of unit digit. This is the answer of series.**

In previous example 10 + 10^{2} + 10^{3} + 10^{4} + 10^{5} we know the constant multiplier is 10 and first digit is also 10 then n = 5. We need to place ‘1’ for each ‘n’ means 11111. After that add ‘0’ in place of unit digit means 111110.

This is our answer.

**Banking Exam Preparation – ADDITION and SUBTRACTION- Part III**

**Example:-**

- 6 + 0.66 + 0.666 + 0.6666

**Solve – **0.6 + 0.66 + 0.666 + 0.6666

= 6(0.1 + 0.11 + 0.111 + 0.1111)

= 6/9(0.9 + 0.99 + 0.999 + 0.9999)

** **= 6/9(9/10 + 99/100 + 999/1000 + 9999/10000)

= 6/9{(1 – 1/10) + (1 – 1/100) + (1 – 1/1000) + (1 – 1/10000)}

= 6/9{4 – (1/10 + 1/100 + 1/1000 + 1/1000)}

= 6/9{4 – (0.1 + 0.01 + 0.001 + 0.0001)}

We need to find solution of geometric series when cm < 1. We know cm = 0.1 or 1/10

= 6/9[4 – {0.1(1 – 0.1^{4})/1 – 0.1}]

= 6/9{4 – 0.1111}

= 6/9(3.889)

=2.5926

Formula (reminder) –

**Common Digit/9(n – Solution of geometric series of 0.1)**

**Common digit/9{n – (10 ^{n}-1)/10^{n} x 9}**

**Shortcut – Solution of geometric series of 10- if constant multiplier is 0.1 or 1/10 and first number is also 0.1 or 1/10 then place ‘1’ for each ‘n’ and add ‘0’ in place of first digit and place Point after ‘0’. This is the answer of series.**

In previous example 1/10 + 1/100 + 1/1000 + 1/1000 or 0.1 + 0.01 + 0.001 + 0.0001 we know the constant multiplier is 0.1 or 1/10 and first digit is also 0.1 or 1/10 then n = 4. We need to place ‘1’ for each ‘n’ means 1111. After that add ‘0’ in place of first digit means 0111 and after this place Point after ‘0’ means 0.1111. This is the answer.

**Banking Exam Preparation – ADDITION and SUBTRACTION- Part III**

Summary –

**If cm ≥ 1 When**

Series 10 + 100 + 1000 + 10000 or 10 + 10^{2} + 10^{3} + 10^{4 }then answer will be 11110.

9 + 99 + 999 + 9999 then answer will be 11110 – 4 = 11106

1 + 11 + 111 + 1111 then answer will be 11106/9 = 1234

4 + 44 + 444 + 4444 then answer will we 11106x*4/9 = 4936

**If cm < 1 When**

Series 0.1 + 0.01 + 0.001 or 1/10 + 1/100 + 1/1000 then answer will be 0.111

0.9 + 0.99 + 0.999 then answer will be 3 – 0.111 = 2.889

0.1 + 0.11 + 0.111 then answer will be 2.889/9 = 0.321

0.6 + 0.66 + 0.666 then answer will be 0.321 x 6 = 1.926

**Banking Exam Preparation – ADDITION and SUBTRACTION- Part III**

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