4, 10, 16, 22,... are the A.P obtained from the 1st differences of a Quadratic Sequence
with a 1st term of 3. Determine the formula for
the sequence & hence its 25th term
plz show step
What is the difference between the numbers in that sequence?
clearly
a = 4, d = 6
term(n) = a + (n-1)d
= 4 + 6(n-1)
= 4 + 6n - 6
= 6n - 2
term(25) = .....
To find the formula for the sequence and determine its 25th term, we will first need to find the common difference and the common ratio for the given arithmetic progression (A.P.).
Step 1: Find the common difference:
We are given that the given arithmetic progression is obtained from the 1st differences of a Quadratic Sequence. The 1st differences between consecutive terms in an arithmetic progression give us the common difference. In this case, the 1st differences are 4, 6, 6. Notice that the differences between the first few 1st differences are the same, indicating a common difference of 6.
Step 2: Determine the first term of the Quadratic Sequence:
We are given that the first term of the arithmetic progression is 3. The first term of the Quadratic Sequence can be obtained by subtracting the square of the common difference from the first term of the arithmetic progression. So, the first term of the Quadratic Sequence is 3 - (6)^2 = 3 - 36 = -33.
Step 3: Find the formula for the Quadratic Sequence:
The formula for a Quadratic Sequence is given by the formula a_n = an^2 + bn + c, where a, b, and c are constants. We already know the first term of the Quadratic Sequence is -33, so we need to find the values of a, b, and c.
We can use the second term of the arithmetic progression to find the value of b. The second term of the arithmetic progression is 10. Substituting n=2 into the formula for the Quadratic Sequence, we get:
a_2 = (2)^2a + 2b + c = 10.
Simplifying the equation, we get:
4a + 2b + c = 10. (Equation 1)
We can use the third term of the arithmetic progression to find the value of c. The third term of the arithmetic progression is 16. Substituting n=3 into the formula for the Quadratic Sequence, we get:
a_3 = (3)^2a + 3b + c = 16.
Simplifying the equation, we get:
9a + 3b + c = 16. (Equation 2)
To solve Equations 1 and 2, we can subtract Equation 1 from Equation 2 to eliminate c:
(9a + 3b + c) - (4a + 2b + c) = 16 - 10.
Simplifying the equation, we get:
5a + b = 6. (Equation 3)
We have two equations (Equations 1 and 3) with two variables (a and b). We can solve these equations simultaneously to find the values of a and b.
Solving Equations 1 and 3, we get:
4a + 2b + c = 10 (Equation 1)
5a + b = 6. (Equation 3)
From Equation 3, we can rearrange it to b = 6 - 5a and substitute it into Equation 1:
4a + 2(6 - 5a) + c = 10.
Simplifying the equation, we get:
4a + 12 - 10a + c = 10,
-6a + c = -2.
Since there are still two variables (a and c), we need one more equation to solve for both variables. We can use the fourth term of the arithmetic progression, which is 22. Substituting n=4 into the formula for the Quadratic Sequence, we get:
a_4 = (4)^2a + 4b + c = 22.
Simplifying the equation, we get:
16a + 4b + c = 22. (Equation 4)
We can subtract Equation 3 from Equation 4 to eliminate b:
(16a + 4b + c) - (5a + b) = 22 - 6.
Simplifying the equation, we get:
11a + 3b + c = 16. (Equation 5)
Now we have three equations (Equations 3, 5, and -6a + c = -2) with three variables (a, b, and c). We can solve these equations simultaneously to find the values of a, b, and c.
Substituting b = 6 - 5a from Equation 3 into Equation 5, we get:
11a + 3(6 - 5a) + c = 16,
11a + 18 - 15a + c = 16,
-4a + c = -2.
Now we have two equations (Equations -6a + c = -2 and -4a + c = -2) with two variables (a and c). Since the equations are the same, this means a = -2/4 = -1/2.
Substituting a = -1/2 into -6a + c = -2, we get:
-6(-1/2) + c = -2,
3 + c = -2,
c = -2 - 3,
c = -5.
Therefore, the formula for the Quadratic Sequence is:
a_n = (-1/2)n^2 + (6 - 5(-1/2))n - 5.
Simplifying the equation, we get:
a_n = (-1/2)n^2 + 6n + 5.
Step 4: Determine the 25th term of the Quadratic Sequence:
Now that we have the formula for the Quadratic Sequence, we can substitute n = 25 to find the 25th term:
a_25 = (-1/2)(25)^2 + 6(25) + 5.
Simplifying the equation, we get:
a_25 = (-1/2)(625) + 150 + 5,
a_25 = -312.5 + 150 + 5,
a_25 = -157.5 + 150,
a_25 = -7.5.
Therefore, the 25th term of the Quadratic Sequence is -7.5.