How to find the eleventh and sixth term of an A. P are 48 and 23. Calculate the first term and the common difference
To find the first term and the common difference of an arithmetic progression (A.P.), we can use the formulas:
nth term (Tn) = a + (n - 1)d
Where Tn represents the nth term, a represents the first term, n represents the position of the term in the A.P., and d represents the common difference.
Given that the eleventh term (T11) is 48 and the sixth term (T6) is 23, we can set up the following equations:
T11 = a + (11 - 1)d
48 = a + 10d
T6 = a + (6 - 1)d
23 = a + 5d
Now we have a system of two equations with two variables. We can solve it by substitution or elimination method.
Using the elimination method, we can subtract the second equation from the first equation to eliminate the "a" term:
48 - 23 = (a + 10d) - (a + 5d)
25 = 5d
d = 25/5
d = 5
Now that we have found the common difference (d) to be 5, we can substitute this value back into one of the equations to find the first term (a).
23 = a + 5(5)
23 = a + 25
a = 23 - 25
a = -2
Therefore, the first term (a) of the A.P. is -2, and the common difference (d) is 5.
To find the first term and the common difference of an arithmetic progression (A.P.), given the eleventh and sixth terms, you can use the following steps:
Step 1: Identify the given information:
The eleventh term (a11) = 48
The sixth term (a6) = 23
Step 2: Finding the common difference (d):
Use the formula for the nth term of an A.P: an = a1 + (n - 1)d
Substitute the values of the given terms to find the common difference:
a11 = a1 + (11 - 1)d
48 = a1 + 10d ---(1)
a6 = a1 + (6 - 1)d
23 = a1 + 5d ---(2)
Step 3: Solving the equations:
We now have two equations with two unknowns (a1 and d).
Subtract equation (2) from equation (1) to eliminate a1:
48 - 23 = (a1 + 10d) - (a1 + 5d)
25 = 5d
Solving for d, divide both sides of the equation by 5:
d = 25 / 5
d = 5
Step 4: Finding the first term (a1):
Substitute the value of d into equation (2) and solve for a1:
23 = a1 + 5(5)
23 = a1 + 25
a1 = 23 - 25
a1 = -2
Therefore, the first term (a1) of the A.P. is -2, and the common difference (d) is 5.
To find the first term and common difference of an arithmetic progression (A.P.) given two terms, you can use the following steps:
Step 1: Identify the given terms
In this case, the eleventh term (T11) of the A.P. is 48, and the sixth term (T6) is 23.
Step 2: Set up the equations
Using the formula for the nth term of an A.P., we can set up two equations:
T11 = a + (11 - 1) * d (Equation 1)
T6 = a + (6 - 1) * d (Equation 2)
where:
T is the term of the A.P.
a is the first term of the A.P.
d is the common difference of the A.P.
Step 3: Solve the equations
Substitute the values from the given terms into the equations and solve for a and d simultaneously.
Using Equation 1:
48 = a + 10d
Using Equation 2:
23 = a + 5d
Step 4: Eliminate a variable
To eliminate variable a, we can subtract Equation 2 from Equation 1:
48 - 23 = (a + 10d) - (a + 5d)
25 = 5d
Step 5: Solve for d
Divide both sides of the equation by 5:
5d/5 = 25/5
d = 5
Step 6: Substitute d into either equation to solve for a
Using Equation 2:
23 = a + 5 * 5
23 = a + 25
a = 23 - 25
a = -2
So, the first term (a) of the A.P. is -2 and the common difference (d) is 5.