a)
1) Let the first term of the AP be a and the common difference be d.
From the information given, we have:
a + a = 14 (since the first term of the sequence is formed by adding corresponding terms of the AP and GP)
a = 7
Also, the second term of the sequence is formed by adding corresponding terms of the AP and GP:
a + 2d + ar = 34
Substitute the values of a and r (given) into the above equation:
7 + 2d + 7*3 = 34
2d + 21 = 27
2d = 6
d = 3
The first term of the AP is 7, the first term of the GP is 3, and the common difference of the AP is 3.
2) To find the sixth term of the sequence:
a + 5d + ar^5 = 7 + 5*3 + 3*3^5
= 7 + 15 + 243
= 265
The sum of the first six terms of the sequence:
S6 = 6/2 * (2a + (6-1)d + a(1 + r + r^2 + r^3 + r^4 + r^5))
= 3 * (14 + 5*3 + 7(1 + 3 + 9 + 27 + 81 + 243))
= 3 * (14 + 15 + 7*364)
= 3 * 2627
= 7881
b)
Let the first term of the GP be a and the common ratio be r.
Given:
ar = 24
a * r^2 = 12(x + 1)
Substitute the value of ar into the second equation:
24r = 12(x + 1)
2r = x + 1
Since r is a whole number and a geometric progression, possible values for r are 1, 2.
If r = 1:
2 * 1 = x + 1
x = 1
If r = 2:
2 * 2 = x + 1
x = 3
Given x = 1:
a = 24 / r
a = 24
Given x = 3:
a = 24 / r
a = 12
Therefore, the first term of the GP is either 24 or 12.
A sequence is formed by adding corresponding terms of an AP and GP. The first, seconds 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 sequence
b)find the second and third terms of a geometric progression 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
1 answer
1 answer