ACTIVITY VI
Find the sum
1. first 5 terms of the sequence 3.-6. 12. ... 2. terms of a geometric sequence where the first term is 16, the last term is 432, and the common ratio is 3.
3. first 100 terms of the sequence 2, 2, 2. 4. first 11 terms of the sequence 7. -7.7.
5. 64. 16. 4. 1...
Find the sum
1. first 5 terms of the sequence 3.-6. 12. ... 2. terms of a geometric sequence where the first term is 16, the last term is 432, and the common ratio is 3.
3. first 100 terms of the sequence 2, 2, 2. 4. first 11 terms of the sequence 7. -7.7.
5. 64. 16. 4. 1...
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Answers & Comments
1. 33
2. 640
3. 200
4. 7
5. 85
to find the sum of the first five terms::
Given:
a1 = 3
a2 = -6
a3 = 12
n = 5
to find r:
r = a2 ÷ a1
r = -6 ÷ 3
r = -2
the formula for the sum of Geometric Series:
sn = a1 (1-r^n) / 1-r
s5 = 3 (1-(-2⁵) / 1-(-2)
s5 = 3 (1-(-32) / 3
s5 = 3 (33) / 3
s5 = 99 / 3
s5 = 33
to find the term of 432:
the general formula for Geometric Sequence:
an = a1(r)^n-1
432 = 16 (3)^n-1
432 / 16 = 16 (3)^n-1 / 16
27 = 3^n-1
3³ = 3^n-1 ( cancel the 3 )
3 = n - 1
4 = n
to find the sum of the first four terms
to formula for the sum of Geometric Series:
sn = a1 (1 - r^n) / 1 - r
sn = 16 (1 - 3⁴) / 1 - 3
sn = 16 (1 - 81) / -2
sn = 16 (-80) / -2
sn = 16 (40)
sn = 640
to find the sum of the first 100 terms:
Given:
a1 = 2
a2 = 2
r = 1
n = 100
since the common ratio for the formula for the sum of Geometric Series can not be 1, we'll use another method for r = 1
sn = (a1 (10))(10)
sn = (2 (10))(10)
sn = (20)(10)
sn = 200
to find the sum of the first 11 terms:
Given
a1 = -7
a2 = 7
n = 11
to find r:
r = a2 ÷ a1
r = 7 ÷ -7
r = -1
the formula for the sum of Geometric Series:
sn = a1 (1-r^n) / 1-r
sn = -7 (1-(-1)¹¹) / 1-(-1)
sn = -7 (1-(-1) / 2
sn = -7 (2) / 2
sn = -7
( I made a mistake in my answer. It should be -7, my apologies. )
to find the sum of terms:
Given
a1 = 64
a2 = 16
a3 = 4
a4 = 1
n = 4
to find r:
r = a2 ÷ a1
r = 16 ÷ 64
r = 1/4
the formula for the sum of Geometric Series:
sn = a1 (1-r^n) / 1-r
sn = 64 (1-(¼)⁴) / 1-¼
sn = 64 (1-(1/256) / ¾
sn = 64 (255/256) / ¾
sn = 64 (85/64)
sn = 85
:D