This document provides information on calculating the sum of terms in a geometric series using formulas. It gives three examples of finding the sum of terms for different geometric sequences: (1) the sum of the first 12 terms of 4, 16, 64,... (2) the sum of the first 7 terms of -1, -5, -25, -125,... (3) the sum of the first 10 terms where the first term is 1/2 and the fourth term is 4. The key steps are to identify the initial term (A1), common ratio (r), and number of terms (n); then apply the formula Sn = A1(1 - rn)/(1 - r) to calculate the sum, where r

1. MS. G. MARTIN

2. To find the sum of the first n terms of a geometric series, we use the following
formulas:
Sn =
A1 (1 – rn )
1 – r
, r ≠ 1 and Sn =
A1 − rXn
1 – r
, r ≠ 1
Where A1 = first term
An = nth term
r = common ratio
Sn = sum of the first n terms

3. Example 1.
Find the sum of the first 12 terms of the geometric sequence 4, 16, 64, …...
Steps Solution
1. Enumerate the given information. Here A1 = 4
2. Find the common ratio using the
formula r =
An
An − 1
r =
16
4
= 4

4. Steps Solution
3. Solve for the sum of the first 12 terms
using the formula
Sn =
A1 (1 – rn )
1 – r
, r ≠ 1
Sn =
A1 (1 – rn )
1 – r
, r ≠ 1
S12 =
4 (1 – 412 )
1 − 4
, r ≠ 1
=
4(1 – 16,777,216)
−3
=
4( –16,777,215 )
−3
= 4(5, 592, 405)
S12 = 22,369,620

5. Example 2.
Find the sum of the first seven terms of the geometric sequence
-1, -5, -25, -125,…..
Steps Solution
1. Enumerate the given information. Here A1 = -1
2. Find the common ratio. r =
−5
−1
= 5

6. Steps Solution
3. Solve for the sum of the first seven
terms using the formula
Sn =
A1 (1 – rn )
1 – r
, r ≠ 1
Sn =
A1 (1 – rn )
1 – r
, r ≠ 1
S7 =
−1 (1 – 57 )
1 – 5
,
S7 =
−1 ( 1 – 78,125 )
−4
S7 =
−1 (–78,124)
−4
=
𝟕𝟖,𝟏𝟐𝟒
−𝟒
S7 = -19,531

7. Example 3. Find the sum of the first ten terms of the geometric sequence
whose first term is
1
2
and whose fourth term is 4.
Steps Solution
1. Enumerate the given information. Here, A1 =
1
2
and A4 = 4
2. Find the common ratio.
If A1 =
1
2
and A4 = 4, we have
An = A1rn-1
A4 = A1r4-1
4 =
1
2
r3
4
1
2
= 4 .
2
1
= 8
8 = r3 ;
3
8 = r
2 = r

8. Steps Solution
3. Solve for the sum of the first ten terms
using the formula
Sn =
A1 (1 – rn )
1 – r
, r ≠ 1
If A1 =
1
2
, A4 = 4, r = 2, n = 10, we have
S10 =
1
2
(1 – 210 )
1 – 2
S10 =
1
2
(1 – 1,024)
−1
S10 =
1(−1,023)
2
÷ -1
S10 =
−1,023
2
÷ -1
S10 =
𝟏,𝟎𝟐𝟑
𝟐

9. Activity

10. Find the needed term of the given geometric sequence
1. 5, 10, 20, 40, …, find S10
Steps Solution
1. Enumerate the given information. A1 = 5 ; n = 10 ; r =
10
5
= 2
2. Solve for the sum of the first ten
terms using the formula
Sn =
A1 (1 – rn )
1 – r
, r ≠ 1
S10 =
5(1 – 210 )
1 – 2
S10 =
5(1 – 1,024 )
−1
S10 =
5(−1,023 )
−1
=
−5,115
−1
S10 = 5,115

11. 2. 960, 240, 60, 15, …, find S5
Steps Solution
1. Enumerate the given information. A1 = 960 ; n = 5 ; r =
240
960
=
1
4
2. Solve for the sum of the first five
terms using the formula
Sn =
A1 (1 – rn )
1 – r
, r ≠ 1
S5 =
960(1 – ( 1
4
)5
1 – 1
4
S5 =
960(1 – 1
1,024
)
3
4
S5 =
𝟓,𝟏𝟏𝟓
𝟒