Sum of n terms of a Geometric Progression

We will learn how to find the sum of n terms of the Geometric Progression {a, ar, ar2, ar3, ar4, ...........}

To prove that the sum of first n terms of the Geometric Progression whose first term β€˜a’ and common ratio β€˜r’ is given by

Sn = a(rnβˆ’1rβˆ’1)

β‡’ Sn = a(1βˆ’rn1βˆ’r), r β‰  1.

Let Sn denote the sum of n terms of the Geometric Progression {a, ar, ar2, ar3, ar4, ...........} with first term β€˜a’ and common ratio r. Then,

Now, the nth terms of the given Geometric Progression = a βˆ™ rnβˆ’1.

Therefore, Sn = a + ar + ar2 + ar3 + ar4 + ............... + arnβˆ’2 + arnβˆ’1 ............ (i)

Multiplying both sides by r, we get,

rSn = ar + ar2 + ar3 + ar4 + ar4 + ................ + arnβˆ’1 + arn ............ (ii)

____________________________________________________________

On subtracting (ii) from (i), we get

Sn - rSn = a - arn

β‡’ Sn(1 - r) = a(1 - rn)

β‡’ Sn = a(1βˆ’rn)(1βˆ’r)

β‡’ Sn = a(rnβˆ’1)(rβˆ’1)

Hence, Sn = a(1βˆ’rn)(1βˆ’r) or, Sn = a(rnβˆ’1)(rβˆ’1)


Notes:

(i) The above formulas do not hold for r = 1. For r = 1, the sum of n terms of the Geometric Progression is Sn = na.

(ii)When the numerical value of r is less than 1 (i.e., - 1 < r < 1), then the formula Sn = a(1βˆ’rn)(1βˆ’r) is used.

(iii) When the numerical value of r is greater than 1 (i.e., r > 1 or, r < -1) then the formula Sn = a(rnβˆ’1)(rβˆ’1) is used.

(iv) When r = 1, then Sn = a + a + a + a + a + .................... to n terms = na.

(v) If l is the last term of the Geometric Progression, then l = arnβˆ’1.

Therefore, Sn = a(1βˆ’rn1βˆ’r) = (aβˆ’arn1βˆ’r) = aβˆ’(arnβˆ’1)r(1βˆ’r) = aβˆ’lr1βˆ’r

Thus, Sn = aβˆ’lr1βˆ’r

Or, Sn = lrβˆ’arβˆ’1, r β‰  1.

 

Solved examples to find the Sum of first n terms of the Geometric Progression:

1. Find the sum of the geometric series:

4 - 12 + 36 - 108 + .............. to 10 terms

Solution:

The first term of the given Geometric Progression = a = 4 and its common ratio = r = βˆ’124 = -3.

Therefore, the sum of the first 10 terms of the geometric series

= a βˆ™ rnβˆ’1rβˆ’1, [Using the formula Sn = a(rnβˆ’1)(rβˆ’1) since, r = - 3 i.e., r < -1]

= 4 βˆ™ (βˆ’3)10βˆ’1βˆ’3βˆ’1

= 4 βˆ™ (βˆ’3)10βˆ’1βˆ’4

= - (-3)10 - 1

= -59048


2. Find the sum of the geometric series:

1 + 12 + 14 + 18 + 116 + .............. to 10 terms

Solution:

The first term of the given Geometric Progression = a = 1 and its common ratio = r = 121 = 12

Therefore, the sum of the first 10 terms of the geometric series

S10 = a(1βˆ’r10)(1βˆ’r)

β‡’ S10 = 1 βˆ™ (1βˆ’(12)10)(1βˆ’12)

β‡’ S10 = 2(1210)

β‡’ S10 = 2(210βˆ’1210)

β‡’ S10 = 2(1024βˆ’11024)

β‡’ S10 = 1024βˆ’1512

β‡’ S10 = 1023512

Note that we have used formula Sn = a((1βˆ’rn)(1βˆ’r) since r = 1/4 i.e., r < 1]

 

3. Find the sum of 12 terms of the Geometric Progression 3, 12, 48, 192, 768, ................

Solution:

The first term of the given Geometric Progression = a = 3 and its common ratio = r = 123 = 4

Therefore, the sum of the first 12 terms of the geometric series

Therefore, S12 = ar12βˆ’1rβˆ’1

= 3(412βˆ’14βˆ’1)

= 3(16777216βˆ’13)

= 16777216 - 1

= 16777215


4. Find the sum to n terms: 5 + 55 + 555 + 5555 + .............

Solution:

We have 5 + 55 + 555 + 5555 + ............. to n terms

= 5[1 + 11 + 111 + 1111 + .............. + to n terms]

= 59[9 + 99 + 999 + 9999 + ................ + to n terms]

= 59[(10 – 1) + (102 - 1) + (103 - 1) + (104 - 1) + .............. + (10n - 1)]

= 59[(10 + 102 + 103 + 104 + ................ + 10n) – ( 1 + 1 + 1 + 1 + ................ + 1)] n times

= 59[10 Γ— (10nβˆ’1)(10βˆ’1) – n]

= 59[109(10n – 1) – n]

= 581[10n+1 – 10 – 9n]

● Geometric Progression




11 and 12 Grade Math 

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