consider the following (k-4) ; (k+1) ;m ; 5k ... the first 3 terms are arithmetic the last 3terms are geometric. .. determine the values of m and k if both are positive integers
clearly the common difference is 5, from the first two terms.
so, m-(k+1) = 5
now, the common ratio tells us that
m/(k+1) = 5k/m
solving both those equations, we get
(k,m) = (4,10)
and the terms are
0,5,10,20
and you can see the AP and GP
You can calculated my question where ,asking like that ,consider the following term (k-4),(k-1),m,5k
To determine the values of m and k, we need to analyze the given sequence.
The first three terms, (k-4), (k+1), and m, are said to be in arithmetic progression because the difference between consecutive terms is constant. Thus, we can write the equation:
(k+1) - (k-4) = m - (k+1)
By simplifying the equation, we get:
k + 1 - k + 4 = m - k - 1
5 = m - k - 1
m - k = 6 -----(Equation 1)
Now, let's focus on the last three terms, m, 5k, and 5k. These terms are in a geometric progression because there is a common ratio between consecutive terms. Hence, we can write:
5k / m = 5k / 5k
By simplifying, we get:
1 / m = 1
This implies that m = 1 -----(Equation 2)
Substituting Equation 2 into Equation 1, we have:
1 - k = 6
-k = 5
k = -5
Since both m and k need to be positive integers, we can conclude that no solution exists for the given conditions.