The difference between the 11th term and 4th term of an A.P is 49 and if the 11th term is 2 24/25 times the fourth term, determine the common difference, the first term of the sequence and the 61st term of the sequence
Why did the arithmetic progression join the circus? Because it wanted to find its common difference! Let's solve this riddle step by step.
Let's assume that the first term of the arithmetic progression is "a" and the common difference is "d". According to our information, the 11th term (a + 10d) is 49 more than the 4th term (a + 3d). We can write this as:
a + 10d - (a + 3d) = 49
Simplifying this equation gives us:
7d = 49
Dividing both sides by 7, we find:
d = 7
Now that we know the common difference, we can find the first term using the information that the 11th term is 2 24/25 times the fourth term. This can be written as:
(a + 10d) = (2 24/25) * (a + 3d)
Simplifying this equation gives us:
9a + 30d = 49 * 25/24
Since we know that d = 7, we can substitute this value:
9a + 30 * 7 = 49 * 25/24
Simplifying further gives us:
9a + 210 = 51.04
Subtracting 210 from both sides:
9a = 51.04 - 210
9a = -158.96
Dividing both sides by 9:
a = -17.66
So, the first term of the sequence is approximately -17.66.
Finally, let's find the 61st term of the sequence. We can use the formula for the nth term of an arithmetic progression:
Tn = a + (n - 1) × d
Substituting a = -17.66 and d = 7, we get:
T61 = -17.66 + (61 - 1) × 7
T61 = -17.66 + 60 × 7
T61 = -17.66 + 420
T61 = 402.34
So, the 61st term of the sequence is approximately 402.34.
I hope this mathematical circus act brought a smile to your face!
To find the common difference and the first term of the arithmetic progression (A.P.), we can use the formula for the nth term of an A.P., which is given by:
Tn = a + (n - 1)d
Where Tn is the nth term, a is the first term, n is the position of the term, and d is the common difference.
Let's use this information to find the common difference and the first term.
1) Find the common difference (d):
Given that the difference between the 11th term and the 4th term is 49, we can set up the equation:
T11 - T4 = 49
Using the formula for the nth term, we substitute the values:
(a + (11 - 1)d) - (a + (4 - 1)d) = 49
Simplifying the equation, we get:
10d - 3d = 49
7d = 49
d = 7
Therefore, the common difference is 7.
2) Find the first term (a):
Given that the 11th term is 2 24/25 times the fourth term, we can set up the equation:
T11 = 2 24/25 * T4
Using the formula for the nth term, we substitute the values:
a + (11 - 1)7 = 2 24/25 * (a + (4 - 1)7)
Simplifying the equation, we get:
a + 60 = 2 24/25 * (a + 21)
Cross-multiplying, we have:
a + 60 = (2 * 25 + 24) / 25 * (a + 21)
a + 60 = 74/25 * (a + 21)
Expanding the equation, we get:
a + 60 = (74a + 1614) / 25
Multiplying both sides by 25 to eliminate the fraction, we have:
25a + 1500 = 74a + 1614
Subtracting 25a and 1614 from both sides, we obtain:
1500 - 1614 = 74a - 25a
-114 = 49a
Dividing both sides by 49, we get:
a = -114 / 49
a = -2.3265 (rounded to four decimal places)
Therefore, the first term rounded to four decimal places is approximately -2.3265.
3) Find the 61st term (T61):
Using the formula for the nth term, we substitute the values:
T61 = a + (61 - 1)7
T61 = a + (60)7
T61 = a + 420
Substituting the value of the first term we found earlier, we have:
T61 = -2.3265 + 420
T61 = 417.6735 (rounded to four decimal places)
Therefore, the 61st term rounded to four decimal places is approximately 417.6735.
In summary:
- The common difference (d) is 7.
- The first term (a) is approximately -2.3265.
- The 61st term (T61) is approximately 417.6735.
To find the solution to this problem, we need to use the formula for the nth term of an arithmetic progression (A.P), which is:
An = A1 + (n-1)d
where An represents the nth term, A1 is the first term, n is the term number, and d is the common difference.
1. Difference between the 11th and 4th term:
According to the problem, the difference between the 11th term (A11) and the 4th term (A4) is 49. So, we can write this as:
A11 - A4 = 49
2. Relationship between the 11th and 4th term:
The problem states that the 11th term is 2 24/25 times the fourth term. Mathematically, we can represent this as:
A11 = (2 24/25) * A4
Using the mixed fraction as an improper fraction, we rewrite the equation as:
A11 = (49/25) * A4
or
25 * A11 = 49 * A4
3. Solving the equations:
Now we have two equations:
A11 - A4 = 49 (equation 1)
25 * A11 = 49 * A4 (equation 2)
To solve these equations and find the common difference (d) and the first term (A1), we can use the method of substitution or elimination.
Let's use substitution to solve the equations:
From equation 1, we can write:
A11 = 49 + A4
Substituting A11 in equation 2, we get:
25 * (49 + A4) = 49 * A4
Expanding and simplifying, we have:
1225 + 25A4 = 49A4
Bringing the terms with A4 on one side, we get:
49A4 - 25A4 = 1225
24A4 = 1225
Dividing both sides by 24, we find:
A4 = 1225/24
Now that we have A4, we can substitute it back into equation 1 to find A11:
A11 - (1225/24) = 49
Multiplying through by 24 to eliminate the fraction, we get:
24A11 - 1225 = 1176
Adding 1225 to both sides, we find:
24A11 = 2401
Dividing both sides by 24, we get:
A11 = 100.0417
4. Finding the common difference (d) and the first term (A1):
Now that we have A4 = 1225/24 and A11 = 100.0417, we can use either of the two equations.
Using equation 1, we can substitute the values of A11 and A4 to find d:
A11 - A4 = 49
100.0417 - (1225/24) = 49
Doing the arithmetic, we find:
100.0417 - 50.2083 = 49
49.8334 = 49
Therefore, the common difference (d) is 49.
Now, to find the first term (A1), we can use equation 1 again:
A1 + 10 * d = A11
A1 + 10 * 49 = 100.0417
A1 + 490 = 100.0417
Subtracting 490 from both sides, we get:
A1 = -389.9583
So, the first term (A1) is approximately -389.9583.
5. Finding the 61st term:
We can use the formula for the nth term of an A.P to find the 61st term (A61):
A61 = A1 + (61-1) * d
Substituting the values we found earlier:
A61 = -389.9583 + 60 * 49
Doing the arithmetic, we get:
A61 = -389.9583 + 2940
A61 = 2550.0417
Therefore, the 61st term (A61) is approximately 2550.0417.
In summary, the solution to the problem is:
- The common difference (d) is 49
- The first term (A1) is approximately -389.9583
- The 61st term (A61) is approximately 2550.0417.
7d = 49, so d = 7
a+10d = 2 24/25 (a+3d) so a = 4
a_61 = 4 + 60*7 = _____