{"id":2488,"date":"2021-10-01T06:46:54","date_gmt":"2021-10-01T06:46:54","guid":{"rendered":"http:\/\/stgwebsite.mindspark.in\/wordpress\/?page_id=2488"},"modified":"2022-01-02T06:30:44","modified_gmt":"2022-01-02T06:30:44","slug":"sum-of-a-geometric-progression","status":"publish","type":"page","link":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/sum-of-a-geometric-progression\/","title":{"rendered":"Sum of a Geometric Progression"},"content":{"rendered":"

[et_pb_section fb_built=”1″ admin_label=”Section” module_class=”mainsec” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#e0f2fd” z_index=”1″ custom_padding=”5px||5px||true|false” locked=”off” collapsed=”off” global_colors_info=”{}”][et_pb_row column_structure=”3_5,2_5″ custom_padding_last_edited=”on|phone” _builder_version=”4.10.8″ _module_preset=”default” background_color=”#FFFFFF” width=”100%” max_width=”1310px” custom_padding=”|51px|40px|51px|false|true” custom_padding_tablet=”” custom_padding_phone=”|40px|30px|40px|false|true” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}”][et_pb_column type=”3_5″ admin_label=”Column L” _builder_version=”4.9.10″ _module_preset=”default” global_colors_info=”{}”][et_pb_text admin_label=”Acute Angles
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Sum of a Geometric Progression<\/strong> <\/span><\/h1>\n

[\/et_pb_text][et_pb_text admin_label=”Text” _builder_version=”4.11.3″ _module_preset=”default” text_font_size=”16px” header_2_font=”|600|||||||” header_2_text_color=”#a01414″ header_3_font=”|600|||||||” custom_padding=”15px|15px||4px|false|false” global_colors_info=”{}”]<\/p>\n

Sum of a Geometric Progression – Definition, formula, and derivation:<\/strong><\/h2>\n

The series in which the ratio of any two consecutive numbers is the same is known as geometric progression. For example, consider a series containing the terms 4, 16, 64, 256, and so on. The ratio of any two consecutive numbers i.e<\/span> 16 <\/span>4<\/span>=<\/span>64 <\/span>16<\/span>=<\/span>256 <\/span>64<\/span>=4<\/span>. <\/span>Similarly, consider a series having 81,27,9,3, and so on. The ratio of any two consecutive numbers is <\/span>1 <\/span>3<\/span><\/p>\n

So, a geometric progression can be expressed in the form of a, ar, <\/span>ar<\/span>2<\/sup><\/span>, ar<\/span>3<\/sup><\/span>, ar<\/span>4<\/sup><\/span>\u2026\u2026ar<\/span>n-1<\/sup><\/span><\/p>\n

Where a = the first term<\/span>
<\/span> r = common ratio<\/span>
<\/span> n = number of terms in the GP<\/span><\/p>\n

Now let us learn how to find the sum of a geometric progression series here.<\/span><\/p>\n

Sum of a geometric progression formula:<\/strong><\/h2>\n

If the geometric progression can be expressed as a, ar, <\/span>ar<\/span>2<\/sup><\/span>, ar<\/span>3<\/sup><\/span>,…ar<\/span>n-1<\/sup><\/span>, then the sum of the geometric progression S<\/span>n<\/span> = a+ar+ar<\/span>2<\/sup><\/span>+ar<\/span>3<\/sup><\/span>+\u2026..+ar<\/span>n-1<\/sup><\/span>.<\/span><\/p>\n

Now the common ratio can either be r = 1 or r>1 or r<1. We know that r\u22600 in a geometric progression.\u00a0<\/span><\/p>\n

If r = 1, then S<\/span>n<\/span> = a+a(1)+a(1)<\/span>2<\/sup><\/span>+a(1)<\/span>3<\/sup><\/span>+….+a(1)<\/span>n-1<\/sup><\/span>\u00a0= na.<\/span><\/p>\n

If r >1, then S<\/span>n<\/span> = (<\/span>a ( <\/span>r<\/span>n<\/span>-1<\/span> ))\/<\/span>r-1<\/span>\u00a0<\/span><\/p>\n

And when r < 1, S<\/span>n<\/span> = <\/span>S = (<\/span>a (1- <\/span>r<\/span>n<\/span> ))\/<\/span>1-r<\/span><\/p>\n

Where,<\/span><\/p>\n

a = first term<\/span><\/p>\n

r = common ratio and\u00a0<\/span><\/p>\n

n = number of terms in the GP<\/span><\/p>\n

A geometric progression that has an infinite number of terms is called an infinite geometric progression. The sum of an infinite geometric series can be calculated using the formula<\/span>
<\/span>S = <\/b>a\/( <\/b>1-r)<\/b> where <\/span>r<\/span>0 and | r | <1<\/span>.<\/span><\/p>\n

Let us now understand how these formulas are derived.\u00a0<\/span><\/p>\n

Derivation of the formulas:<\/strong><\/h2>\n

The sum of a GP can be expressed in the form as <\/span>S<\/span>n<\/span> = a+ar+ar<\/span>2<\/sup><\/span>+ar<\/span>3<\/sup><\/span>+\u2026..+ar<\/span>n-1<\/sup><\/span>\u00a0(1)<\/span><\/p>\n

Multiply Equation (1) by the common ratio r, we get<\/span><\/p>\n

Sr = ar+<\/span>ar<\/span>2<\/sup><\/span>+ ar<\/span>3<\/sup><\/span>+ ar<\/span>4<\/sup><\/span>\u2026\u2026+ar<\/span>n<\/sup><\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0– Eqn (2)\u00a0<\/span><\/p>\n

Subtracting Eqn (2) from Eqn (1), we get<\/span><\/p>\n

S-Sr = (a+ar+ar<\/span>2<\/sup><\/span>+ar<\/span>3<\/sup><\/span>+\u2026..+arn-1<\/sup><\/span>) – (<\/span>ar+<\/span>ar<\/span>2<\/sup><\/span>+ ar<\/span>3<\/sup><\/span>+ ar<\/span>4<\/sup><\/span>\u2026\u2026+ar<\/span>n<\/sup><\/span> )<\/span><\/p>\n

(1-r) S = a (1- r<\/span>n<\/sup><\/span>)<\/span><\/p>\n

S = (<\/span>a (1- <\/span>r<\/span>n<\/span> ))\/<\/span>1-r<\/span> ( <\/span>where r <1)<\/span><\/p>\n

When r >1,\u00a0 S = (<\/span>a ( <\/span>r<\/span>n<\/span>-1<\/span> ))\/<\/span>r-1<\/span>\u00a0<\/span><\/p>\n

Now, what happens in the case of an infinite geometric progression? In that case, since the value of n is not fixed and it tends to infinity, how do we calculate the sum of the geometric progression?<\/span><\/p>\n

Let us understand what a convergent series and what a divergent series are before we dive into the derivation of the sum of an infinite geometric progression.<\/span><\/p>\n

An infinite series is said to be a <\/span>convergent series<\/b> if the sum approaches a finite number.<\/span><\/p>\n

A <\/span>divergent series<\/b> is an infinite series that is not convergent. An infinite series where the numbers do not approach zero is diverging.<\/span><\/p>\n

When the common ratio in a geometric progression is greater than 1 or less than -1, the geometric progression is a divergent series. The sum of an infinite divergent series can not be determined and it tends to infinity.\u00a0<\/span><\/p>\n

The sum of an infinite geometric progression can be calculated only if it is a convergent series.<\/span><\/p>\n

We know that when r <1, the sum of the geometric progression is <\/span>S = <\/span>a (1- <\/span>r<\/span>n<\/span> )<\/span>1-r<\/span>.<\/span><\/p>\n

In a geometric progression where n \u2192\u221e,<\/span><\/p>\n

S = (<\/span>a (1- <\/span>r<\/span>n<\/span> ))\/<\/span>1-r<\/span>\u00a0<\/span><\/p>\n

I.e S =( <\/span>a \/(<\/span>1-r)) <\/span>– (<\/span>a <\/span>r<\/span>n<\/sup>\/(<\/span>1-r))<\/span><\/p>\n

Let us consider a GP with a common ratio of 0.5 and <\/span>a\u00a0 <\/span>r<\/span>n<\/span>1-r<\/span> part of the above equation. Now, if we are to calculate the sum of the first 10 terms, 0.5<\/span>10<\/span> = 0.00097 and for the first 20 terms, 0.5<\/span>20<\/span> = 0.00000095. So, we can say that when the terms increase, the value of r<\/span>n<\/span> will approach zero and when n \u2192 \u221e, the value of <\/span>a\u00a0 <\/span>r<\/span>n<\/sup>\/(<\/span>1-r)<\/span>\u2192 <\/span>0 when <\/span>r<\/span>0 and | r | <1<\/span>.<\/span><\/p>\n

Thus, S = <\/b>a\/(<\/b>1-r)<\/b><\/p>\n

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Explore Other Topics<\/h1>\n

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\n
Geometry<\/a><\/div>\n
Trigonometry<\/a><\/div>\n
Operations<\/a><\/div>\n
Numbers<\/a><\/div>\n<\/div>\n

[\/et_pb_text][et_pb_text admin_label=”Related Concepts
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Related Concepts<\/h1>\n

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\n
Geometric Progression<\/a><\/div>\n
AP and GP<\/a><\/div>\n
Sum of infinite GP<\/a><\/a><\/div>\n<\/div>\n

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Frequently Asked Questions\u00a0<\/span><\/span><\/h1>\n

1.<\/strong> In which GP does the sum of an infinite number of terms exist?<\/span><\/p>\n

Ans:<\/strong> The sum of infinite terms can be calculated in a Geometric progression if it is a convergent series i.e. where the common ratio is between -1 and +1 and r \u22600.,\u00a0<\/span><\/p>\n

2.<\/strong> What is the condition for Geometric Progression:<\/span><\/p>\n

Ans:<\/strong> The ratio between any two successive terms in the series is constant and is known as the common ratio. The reciprocal of the terms in the geometric progression also form a geometric progression.\u00a0<\/span><\/p>\n

3.<\/strong> What is the formula for the sum of an infinite GP?<\/span><\/p>\n

Ans:<\/strong> We can calculate the sum of an infinite GP using the formula<\/span><\/p>\n

S<\/span>\u221e<\/span> = <\/span>\u00a0<\/b>a \/ <\/b>1-r<\/b>where <\/span>r<\/span>0 and | r | <1<\/span>.<\/span><\/p>\n

Where a =\u00a0 first term and r = common ratio<\/span><\/p>\n

4.<\/strong> How to find the sum of n terms of a geometric series?<\/span><\/p>\n

Ans:<\/strong> The sum of n terms of a GP can be calculated using the formula\u00a0<\/span><\/p>\n

S<\/span>n<\/span> = <\/span>a \/((1- <\/span>r<\/span>n<\/span> )<\/span>1-r<\/span> ) ( <\/span>where r <1)<\/span><\/p>\n

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