Emmanuel Baah
Apr 24
In an exponential sequence the 6th term is 8 time the 3rd term and the sum of the 7th and 8th term is 192.find 1.the common ratio 2.the first term 3.the sum of the 5th to 11th term inclusive?
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ChatGPT
Apr 24
Let the first term of the exponential sequence be a and the common ratio be r.
Given that the 6th term is 8 times the 3rd term, we have:
a * r^5 = 8 * a * r^2
r^3 = 8
r = 2
Given that the sum of the 7th and 8th term is 192, we have:
a * r^6 + a * r^7 = 192
a * 2^6 + a * 2^7 = 192
64a + 128a = 192
192a = 192
a = 1
Therefore, the common ratio is 2 and the first term is 1.
To find the sum of the 5th to 11th term inclusive, we can use the formula for the sum of n terms of an exponential sequence:
S_n = a * (r^n - 1) / (r - 1)
Sum of the 5th to 11th term inclusive:
S_11 - S_4 = a * (r^11 - 1) / (r - 1) - a * (r^4 - 1) / (r - 1)
= 1 * (2^11 - 1) / (2 - 1) - 1 * (2^4 - 1) / (2 - 1)
= 2047 - 15
= 2032
Therefore, the sum of the 5th to 11th term inclusive is 2032.
Given that the 6th term is 8 times the 3rd term, we have:
a * r^5 = 8 * a * r^2
r^3 = 8
r = 2
Given that the sum of the 7th and 8th term is 192, we have:
a * r^6 + a * r^7 = 192
a * 2^6 + a * 2^7 = 192
64a + 128a = 192
192a = 192
a = 1
Therefore, the common ratio is 2 and the first term is 1.
To find the sum of the 5th to 11th term inclusive, we can use the formula for the sum of n terms of an exponential sequence:
S_n = a * (r^n - 1) / (r - 1)
Sum of the 5th to 11th term inclusive:
S_11 - S_4 = a * (r^11 - 1) / (r - 1) - a * (r^4 - 1) / (r - 1)
= 1 * (2^11 - 1) / (2 - 1) - 1 * (2^4 - 1) / (2 - 1)
= 2047 - 15
= 2032
Therefore, the sum of the 5th to 11th term inclusive is 2032.
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Google Gemini AI
Apr 24
1. Common ratio (r):
Given that the 6th term is 8 times the 3rd term, we can write:
6th term = (8 x 3rd term)
Using the formula for the nth term of an exponential sequence:
nth term = a - r^(n-1)
where 'a' is the first term and 'r' is the common ratio.
For the 6th and 3rd terms, we have:
6th term = a - r^(6-1) = a - r^5
3rd term = a - r^(3-1) = a - r^2
Substituting into the given equation, we get:
a - r^5 = 8 - a - r^2
Dividing both sides by a - r^2, we get:
r^3 = 8
Taking the cube root of both sides, we find the common ratio:
r = 2
2. First term (a):
We don't have enough information directly to find the first term. However, since we know the common ratio, we can use the formula for the nth term to find 'a'.
Given that the sum of the 7th and 8th terms is 192, we can write:
7th term + 8th term = 192
Using the formula for the nth term, we have:
7th term = a - r^(7-1) = a - r^6
8th term = a - r^(8-1) = a - r^7
Substituting into the given equation, we get:
a - r^6 + a - r^7 = 192
a(r^6 + r^7) = 192
Substituting r = 2, we get:
a(2^6 + 2^7) = 192
a(64 + 128) = 192
a(192) = 192
Therefore, the first term is a = 1.
3. Sum of the 5th to 11th terms inclusive:
Using the formula for the sum of n terms of an exponential sequence:
Sum = a - (1 - r^n) / (1 - r)
Substituting a = 1, r = 2, and n = 7 (11th term - 5th term + 1), we get:
Sum = 1 - (1 - 2^7) / (1 - 2)
Sum = 1 - (1 - 128) / (-1)
Sum = 1 - (-127) / (-1)
Sum = 127
Given that the 6th term is 8 times the 3rd term, we can write:
6th term = (8 x 3rd term)
Using the formula for the nth term of an exponential sequence:
nth term = a - r^(n-1)
where 'a' is the first term and 'r' is the common ratio.
For the 6th and 3rd terms, we have:
6th term = a - r^(6-1) = a - r^5
3rd term = a - r^(3-1) = a - r^2
Substituting into the given equation, we get:
a - r^5 = 8 - a - r^2
Dividing both sides by a - r^2, we get:
r^3 = 8
Taking the cube root of both sides, we find the common ratio:
r = 2
2. First term (a):
We don't have enough information directly to find the first term. However, since we know the common ratio, we can use the formula for the nth term to find 'a'.
Given that the sum of the 7th and 8th terms is 192, we can write:
7th term + 8th term = 192
Using the formula for the nth term, we have:
7th term = a - r^(7-1) = a - r^6
8th term = a - r^(8-1) = a - r^7
Substituting into the given equation, we get:
a - r^6 + a - r^7 = 192
a(r^6 + r^7) = 192
Substituting r = 2, we get:
a(2^6 + 2^7) = 192
a(64 + 128) = 192
a(192) = 192
Therefore, the first term is a = 1.
3. Sum of the 5th to 11th terms inclusive:
Using the formula for the sum of n terms of an exponential sequence:
Sum = a - (1 - r^n) / (1 - r)
Substituting a = 1, r = 2, and n = 7 (11th term - 5th term + 1), we get:
Sum = 1 - (1 - 2^7) / (1 - 2)
Sum = 1 - (1 - 128) / (-1)
Sum = 1 - (-127) / (-1)
Sum = 127
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