The 2nd, 5th, and 14th term Of an AP are consecutive terms of a GP. Given that the sum of the three terms of the AP is 9, find the
A, first term and the common difference of the AP.
B common ratio of the GP
C sum of the first three terms of the GP
Let's start by finding the values of the AP terms.
Let the first term of the AP be "a" and the common difference be "d".
The 2nd term would then be a + d.
The 5th term would be a + 4d.
The 14th term would be a + 13d.
Now, we know that these three terms are consecutive terms of a geometric progression (GP).
In a geometric progression, each term is obtained by multiplying the previous term by a constant factor called the common ratio (r).
Thus, we have the following equations:
(a + d) * r = a + 4d --(1)
(a + 4d) * r = a + 13d --(2)
Now, we know that the sum of the three terms of the AP is 9:
(a) + (a + d) + (a + 4d) = 9
3a + 5d = 9 --(3)
To solve these equations, we will use the substitution method.
Substituting equation (2) into equation (1) and simplifying, we get:
[(a + 4d) * r] = (a + 13d)
ar + 4dr = a + 13d
ar = a - 9d --(4)
Substituting equation (4) into equation (3), we get:
3a + 5d = 9 (equation from step 3)
3(a - 9d) + 5d = 9
3a - 27d + 5d = 9
3a - 22d = 9 --(5)
Now we have two equations: (4) and (5).
Let's solve these equations simultaneously:
From equation (4): ar = a - 9d
Rewriting this as ar - a = -9d
a(r - 1) = -9d
a = -9d / (r - 1) --(6)
Substitute the value of a from equation (6) into equation (5):
3(-9d / (r - 1)) - 22d = 9
-27d / (r - 1) - 22d = 9
(-27d - 22d(r - 1)) / (r - 1) = 9
Multiplying both sides by (r - 1), we get:
-27d - 22d(r - 1) = 9(r - 1)
-27d - 22dr + 22d = 9r - 9
Now, let's simplify:
-22dr - 27d + 22d = 9r - 9
-22dr - 5d = 9r - 9 - 22d
-22dr - 5d - 9r + 22d = -9
Rearranging the terms:
(-22d - 9r) + (-5d + 22d) = -9
Combining like terms:
-22d - 9r + 17d = -9
Rearranging the terms again:
-22d + 17d - 9r = -9
-5d - 9r = -9
Now, we can divide both sides by -1 to change the sign of the equation:
5d + 9r = 9 --(7)
We have two equations: (6) and (7).
We can solve these equations to find the values of d and r.
To solve this problem, we can use the formulas for both an arithmetic progression (AP) and a geometric progression (GP).
Let's start with the AP terms. The general formula for the nth term of an AP is:
an = a + (n-1)d
Where 'an' represents the nth term, 'a' represents the first term, 'n' represents the position of the term, and 'd' represents the common difference.
Given that the 2nd, 5th, and 14th terms are consecutive terms of a GP, we can set up the following equations:
a + (2-1)d = a + d
a + (5-1)d = a + 4d
a + (14-1)d = a + 13d
We are also given that the sum of the three terms of the AP is 9, therefore:
(a + d) + (a + 4d) + (a + 13d) = 9
3a + 18d = 9
Now we have a system of two equations with two unknowns.
Let's solve this system step-by-step:
1. From the second equation, we get: a + 4d = a + 13d - 9d
Simplifying, we have: 4d = 4d - 9d
Combining like terms, we find: 4d = -5d
Dividing both sides by d, we get: 4 = -5
This means that there is no unique solution for the common difference 'd', and therefore the AP is not well-defined.
Since we didn't find a solution for the common difference 'd', we cannot proceed to find the first term or the common ratio of the GP or the sum of the first three terms of the GP.
please more explanations from to me to get the understanding like solving all
(a+4d)/(a+d) = (a+13d)/(a+4d)
a+d + a+4d + a+13d = 9
AP: a=3/13, d=6/13
T2 = 9/13
T5 = 27/13
T14 = 81/13
Clearly the have a common ratio of 3