A sequence is formed by adding corresponding terms of an AP and GP. The first,second and third terms of the sequence formed are 14,34 and 78 respectively.
a.Given that the common ratio of the GP is 3;
1)find the first term of the AP and GP and the common difference of the AP.
2)Find the sixth term and the sum of the first six terms of the sequece.
b)The second and third terms of a geometric prograssion are 24 and 12(x+1) respectively.Find the whole number value of x and hence the first term given the sum of the first three terms of the progression is 76.
a)
1) Let the first term of the AP be a, the common difference be d, and the first term of the GP be b.
Given that the first three terms of the sequence are 14, 34, and 78:
14 = a + b
34 = a + 3b
78 = a + 9b
Solving these equations simultaneously, we get: a = -2, b = 16, d = 18.
2) The sixth term of the sequence can be found by adding the corresponding terms of the AP and the GP:
6th term = a + 5d + b * 3^5 = -2 + 5*18 + 16*243 = 3894
The sum of the first six terms can be calculated as follows:
Sum of first 6 terms = 6/2 * (2a + 5d + 3(3^5)b) = 3(-2 + 5*18 + 16*243) = 11682
b)
Given that the second and third terms of the geometric progression are 24 and 12(x+1) respectively:
24 = ar
12(x+1) = ar^2
Dividing the second equation by the first, we get:
12(x+1) = 24r
x + 1 = 2r
Given that the sum of the first three terms of the progression is 76:
a + ar + ar^2 = 76
Substitute for r in terms of x from above:
a + 24 + 12(x+1) = 76
a + 24 + 12x + 12 = 76
a + 12x + 36 = 76
a + 12x = 40
We also know that x + 1 = 2r
x = 2r - 1
40 = a + 12(2r - 1)
40 = a + 24r - 12
a = 52 - 24r
Substitute back into x = 2r - 1:
52 - 24r + 12 = 40
12 - 24r = -12
24r = 24
r = 1
Substitute back into a = 52 - 24r:
a = 52 - 24 = 28
Therefore, the value of x is 0 and the first term of the progression is 28.
You say that The sum of the first six terms can be calculated as follows:
Sum of first 6 terms = 6/2 * (2a + 5d + 3(3^5)b)
What makes you think that the resulting sequence is an AP?
You are correct, I apologize for the confusion. The given sequence is not necessarily an arithmetic progression (AP) because it is formed by adding corresponding terms of an arithmetic progression (AP) and a geometric progression (GP).
To find the sum of the the first six terms of the sequence, we need to calculate the sum by summing up the corresponding terms of the AP and GP.
Sum of first 6 terms = (a + b) + (a + 3b) + (a + 9b) + (a + 27b) + (a + 81b) + (a + 243b)
= 6a + (1 + 3 + 9 + 27 + 81 + 243)b
= 6a + 364b
Substitute the values of a = -2 and b = 16 into the equation to find the sum of the first six terms:
Sum of first 6 terms = 6*(-2) + 364*16
= -12 + 5824
= 5812
Therefore, the sum of the first six terms of the sequence is 5812.