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A sequence is said to be a Geometric Progression (GP), if each term (except the first) is obtained by multiplying its preceding term by a non-zero constant.

This constant is called the common ratio and is obtained by dividing any term (except the first term) by its preceding term. It is denoted by r.

Let a_{1},a_{2},a_{3},a_{4},a_{5},.... be a GP.

a_{1} = a and common ratio = r

Then a_{2} = a Ã— r = ar

a_{3} = a_{2} Ã— r = ar Ã— r = ar^{2}

a_{4} = a_{3} Ã— r = ar^{2} Ã— r = ar^{3}

Likewise, a_{n} = ra_{n-1}

a_{n} is called the general term of the GP.

Any required term in the sequence can be found without actually finding the preceding terms.

Thus, the general form of a GP is a,ar, ar^{2}, ar^{3}, ar^{4},....

First term = a = a Ã— 1 = a Ã— r^{0} = a Ã— r^{1-1}

Second term = ar = a Ã— r = a Ã— r^{1} = a Ã— r^{2-1} = ar^{2-1}

Third term = ar^{2} = a Ã— r^{2} = a Ã— r^{3-1} = ar^{3-1}

Fourth term = ar^{3} = a Ã— r^{3} = a Ã— r^{4-1} = ar^{4-1}

â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. and so on

n^{th} term = a_{n} = ar^{n - 1}

Here, the common ratio, â€˜r,â€™ is defined as the ratio of any term (except the first) to its preceding term,

and is expressed as r = $\frac{\text{a}{\text{r}}^{\text{n - 1}}}{\text{a}{\text{r}}^{\text{n - 2}}}$ ; âˆ€ n > 1.

A finite GP â†’ a,ar, ar^{2}, ar^{3}, ar^{4},....,ar^{n-1}

An infinite GP â†’ a, ar ,ar^{2}, ar^{3},....

Correspondingly, a + ar + ar^{2} + ar^{3} + ar^{4} +....+ ar^{n-1} is called a finite GP series,

whereas a + ar + ar^{2} + ar^{3} + ar^{4} +.... is called an infinite GP series.

**Sum of â€˜nâ€™ terms of a G.P.**

Consider a GP with n terms, a,ar, ar^{2}, ar^{3}, ar^{4},....,ar^{n-1}

a + ar + ar^{2} + ar^{3} + ar^{4} +....+ ar^{n-1} is called a finite GP series.

S_{n} = a + ar + ar^{2} + ar^{3} + ar^{4} +....+ ar^{n-1} â€¦(i)

While finding the value of S_{n}, two cases arise.

**Case 1.** When r = 1, by (i), we have S_{n} = a + a + a + a + ... + a (n terms) = na

**Case 2.** When r â‰ 1

Multiplying both the sides of (i) by r, we get

rS_{n} = ar + ar^{2} + ar^{3} + ar^{4} +....+ ar^{n-1} + ar^{n}. (ii)

Subtracting (ii) from (i), we have

S_{n} â€“ rS_{n} = (a + ar + ar^{2} + ar^{3} + ar^{4} +....+ ar^{n-1} ) â€“ (ar + ar^{2} + ar^{3} + ar^{4} +....+ ar^{n-1} + ar^{n})

= a - ar^{n}

â‡’ (1 - r)S_{n} = a(1 - r^{n})

â‡’ S_{n} = $\frac{\text{a(1 -}{\text{r}}^{\text{n}}\text{)}}{\text{1 - r}}$ ; if r < 1

or $\frac{\text{a(1 -}{\text{r}}^{\text{n}}\text{)}}{\text{1 - r}}$ ; if r > 1

The general term, t_{n}, of a GP is given by t_{n} = S_{n} - S_{n-1}; where S_{n} denotes the sum of the first n terms and S_{n-1} denotes the sum of the first (n-1) terms of the GP.

**Simple tricks to find the terms of a GP when their sum is given:**

When we have to find an odd number of terms in a GP whose sum is given, then it is convenient to take the middle term as a and the common ratio as r.

Thus, three terms are taken as: a/r, a, ar and five terms are taken as a/r^{2}, a/r , a, ar, ar^{2}

When we have to find an even number of terms in a GP whose sum is given, we take a/r and ar as the two middle terms and r^{2} as the common difference.

Thus, four terms are taken as: a/r^{3}, a/r, ar, ar^{3} and six terms are taken as: a/r^{5} , a/r^{3}, a/r, ar, ar^{3}, ar^{5}.

This constant is called the common ratio and is obtained by dividing any term (except the first term) by its preceding term. It is denoted by r.

Let a

a

Then a

a

a

Likewise, a

a

Any required term in the sequence can be found without actually finding the preceding terms.

Thus, the general form of a GP is a,ar, ar

First term = a = a Ã— 1 = a Ã— r

Second term = ar = a Ã— r = a Ã— r

Third term = ar

Fourth term = ar

â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. and so on

n

Here, the common ratio, â€˜r,â€™ is defined as the ratio of any term (except the first) to its preceding term,

and is expressed as r = $\frac{\text{a}{\text{r}}^{\text{n - 1}}}{\text{a}{\text{r}}^{\text{n - 2}}}$ ; âˆ€ n > 1.

A finite GP â†’ a,ar, ar

An infinite GP â†’ a, ar ,ar

Correspondingly, a + ar + ar

whereas a + ar + ar

Consider a GP with n terms, a,ar, ar

a + ar + ar

S

While finding the value of S

Multiplying both the sides of (i) by r, we get

rS

Subtracting (ii) from (i), we have

S

= a - ar

â‡’ (1 - r)S

â‡’ S

or $\frac{\text{a(1 -}{\text{r}}^{\text{n}}\text{)}}{\text{1 - r}}$ ; if r > 1

The general term, t

When we have to find an odd number of terms in a GP whose sum is given, then it is convenient to take the middle term as a and the common ratio as r.

Thus, three terms are taken as: a/r, a, ar and five terms are taken as a/r

When we have to find an even number of terms in a GP whose sum is given, we take a/r and ar as the two middle terms and r

Thus, four terms are taken as: a/r

A sequence is said to be a Geometric Progression (GP), if each term (except the first) is obtained by multiplying its preceding term by a non-zero constant.

This constant is called the common ratio and is obtained by dividing any term (except the first term) by its preceding term. It is denoted by r.

Let a_{1},a_{2},a_{3},a_{4},a_{5},.... be a GP.

a_{1} = a and common ratio = r

Then a_{2} = a Ã— r = ar

a_{3} = a_{2} Ã— r = ar Ã— r = ar^{2}

a_{4} = a_{3} Ã— r = ar^{2} Ã— r = ar^{3}

Likewise, a_{n} = ra_{n-1}

a_{n} is called the general term of the GP.

Any required term in the sequence can be found without actually finding the preceding terms.

Thus, the general form of a GP is a,ar, ar^{2}, ar^{3}, ar^{4},....

First term = a = a Ã— 1 = a Ã— r^{0} = a Ã— r^{1-1}

Second term = ar = a Ã— r = a Ã— r^{1} = a Ã— r^{2-1} = ar^{2-1}

Third term = ar^{2} = a Ã— r^{2} = a Ã— r^{3-1} = ar^{3-1}

Fourth term = ar^{3} = a Ã— r^{3} = a Ã— r^{4-1} = ar^{4-1}

â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. and so on

n^{th} term = a_{n} = ar^{n - 1}

Here, the common ratio, â€˜r,â€™ is defined as the ratio of any term (except the first) to its preceding term,

and is expressed as r = $\frac{\text{a}{\text{r}}^{\text{n - 1}}}{\text{a}{\text{r}}^{\text{n - 2}}}$ ; âˆ€ n > 1.

A finite GP â†’ a,ar, ar^{2}, ar^{3}, ar^{4},....,ar^{n-1}

An infinite GP â†’ a, ar ,ar^{2}, ar^{3},....

Correspondingly, a + ar + ar^{2} + ar^{3} + ar^{4} +....+ ar^{n-1} is called a finite GP series,

whereas a + ar + ar^{2} + ar^{3} + ar^{4} +.... is called an infinite GP series.

**Sum of â€˜nâ€™ terms of a G.P.**

Consider a GP with n terms, a,ar, ar^{2}, ar^{3}, ar^{4},....,ar^{n-1}

a + ar + ar^{2} + ar^{3} + ar^{4} +....+ ar^{n-1} is called a finite GP series.

S_{n} = a + ar + ar^{2} + ar^{3} + ar^{4} +....+ ar^{n-1} â€¦(i)

While finding the value of S_{n}, two cases arise.

**Case 1.** When r = 1, by (i), we have S_{n} = a + a + a + a + ... + a (n terms) = na

**Case 2.** When r â‰ 1

Multiplying both the sides of (i) by r, we get

rS_{n} = ar + ar^{2} + ar^{3} + ar^{4} +....+ ar^{n-1} + ar^{n}. (ii)

Subtracting (ii) from (i), we have

S_{n} â€“ rS_{n} = (a + ar + ar^{2} + ar^{3} + ar^{4} +....+ ar^{n-1} ) â€“ (ar + ar^{2} + ar^{3} + ar^{4} +....+ ar^{n-1} + ar^{n})

= a - ar^{n}

â‡’ (1 - r)S_{n} = a(1 - r^{n})

â‡’ S_{n} = $\frac{\text{a(1 -}{\text{r}}^{\text{n}}\text{)}}{\text{1 - r}}$ ; if r < 1

or $\frac{\text{a(1 -}{\text{r}}^{\text{n}}\text{)}}{\text{1 - r}}$ ; if r > 1

The general term, t_{n}, of a GP is given by t_{n} = S_{n} - S_{n-1}; where S_{n} denotes the sum of the first n terms and S_{n-1} denotes the sum of the first (n-1) terms of the GP.

**Simple tricks to find the terms of a GP when their sum is given:**

When we have to find an odd number of terms in a GP whose sum is given, then it is convenient to take the middle term as a and the common ratio as r.

Thus, three terms are taken as: a/r, a, ar and five terms are taken as a/r^{2}, a/r , a, ar, ar^{2}

When we have to find an even number of terms in a GP whose sum is given, we take a/r and ar as the two middle terms and r^{2} as the common difference.

Thus, four terms are taken as: a/r^{3}, a/r, ar, ar^{3} and six terms are taken as: a/r^{5} , a/r^{3}, a/r, ar, ar^{3}, ar^{5}.

This constant is called the common ratio and is obtained by dividing any term (except the first term) by its preceding term. It is denoted by r.

Let a

a

Then a

a

a

Likewise, a

a

Any required term in the sequence can be found without actually finding the preceding terms.

Thus, the general form of a GP is a,ar, ar

First term = a = a Ã— 1 = a Ã— r

Second term = ar = a Ã— r = a Ã— r

Third term = ar

Fourth term = ar

â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦. and so on

n

Here, the common ratio, â€˜r,â€™ is defined as the ratio of any term (except the first) to its preceding term,

and is expressed as r = $\frac{\text{a}{\text{r}}^{\text{n - 1}}}{\text{a}{\text{r}}^{\text{n - 2}}}$ ; âˆ€ n > 1.

A finite GP â†’ a,ar, ar

An infinite GP â†’ a, ar ,ar

Correspondingly, a + ar + ar

whereas a + ar + ar

Consider a GP with n terms, a,ar, ar

a + ar + ar

S

While finding the value of S

Multiplying both the sides of (i) by r, we get

rS

Subtracting (ii) from (i), we have

S

= a - ar

â‡’ (1 - r)S

â‡’ S

or $\frac{\text{a(1 -}{\text{r}}^{\text{n}}\text{)}}{\text{1 - r}}$ ; if r > 1

The general term, t

When we have to find an odd number of terms in a GP whose sum is given, then it is convenient to take the middle term as a and the common ratio as r.

Thus, three terms are taken as: a/r, a, ar and five terms are taken as a/r

When we have to find an even number of terms in a GP whose sum is given, we take a/r and ar as the two middle terms and r

Thus, four terms are taken as: a/r