THE 1ST, 3RD AND 9TH TERM OF A LINEAR SEQUENCE A.P ARE THE 1ST THREE TIMES OF G.P. IF THE 7TH TERM OF THE LINEAR SEQUENCE IS 14. CALCULATE (A) 20TH TERM OF THE LINEAR SEQUENCE (B) SUM OF THE 1ST, 12TH TERM OF THE G.P
To solve this problem, we need to find the common difference of the linear sequence and the common ratio of the geometric progression (G.P). Once we have these values, we can use the formulas for A.P and G.P to find the solutions.
Step 1: Find the common difference of the linear sequence
Let the first term of the linear sequence be a, and the common difference be d. Since the 1st, 3rd, and 9th terms are the first three terms of the G.P, we have the following equations:
a + 2d = a * r
a + 8d = a * r^2
Here, r represents the common ratio of the G.P.
Simplifying these equations, we get:
2d = a * (r - 1)
8d = a * (r^2 - 1)
Step 2: Find the value of r (common ratio)
Dividing the second equation by the first equation, we get:
8d / 2d = (a * (r^2 - 1)) / (a * (r - 1))
Simplifying, we arrive at:
4 = (r + 1)
From this, we can conclude that r = 3.
Step 3: Find the value of d (common difference)
Using the value of r, we can substitute it back into one of the equations. Let's use the first equation:
2d = a * (3 - 1)
2d = 2a
d = a
Step 4: Find the value of a (first term of the linear sequence)
We are given that the 7th term of the linear sequence is 14.
Using the formula for A.P, we have:
a + 6d = 14
Substituting d = a, we get:
a + 6a = 14
7a = 14
a = 2
Step 5: Calculate the 20th term of the linear sequence (A)
Using the formula for A.P, we can find the 20th term:
20th term = a + (n-1)d
= 2 + (20-1)2
= 2 + 19(2)
= 2 + 38
= 40
So, the 20th term of the linear sequence is 40.
Step 6: Calculate the sum of the 1st and 12th term of the G.P (B)
The sum of the 1st n terms of a G.P can be found using the formula:
Sum(n) = a * (r^n - 1) / (r - 1)
The 1st term of the G.P (a) is twice the 1st term of the linear sequence, which is 4.
Using the formula, we can calculate the sum of the 1st and 12th term:
Sum(12) = 4 * (3^12 - 1) / (3 - 1)
= 4 * (531442 - 1) / 2
= 2125768 - 2
= 2125766
So, the sum of the 1st and 12th term of the G.P is 2125766.
In conclusion:
(A) The 20th term of the linear sequence is 40.
(B) The sum of the 1st and 12th term of the G.P is 2125766.
a + 6d = 14
a+2d = ar
a+8d = ar^2
Solve for a,r,d and then figure
A) a+19d
B) a + ar^11