if the sum of the 6th and 7th terms of an arithmetic progression is 60 and the 3rd term is -5 find the common difference
Is 0 know the common difference
To find the common difference in an arithmetic progression, we can use the formula:
Tn = a + (n-1)d
where:
Tn is the nth term of the arithmetic progression
a is the first term of the arithmetic progression
d is the common difference between terms
n is the position of the term in the sequence
We are given the following information:
T3 = -5 (the third term is -5)
T6 + T7 = 60 (the sum of the 6th and 7th terms is 60)
Let's solve for the common difference step-by-step:
Step 1: Find the 6th term (T6)
T6 = a + (6-1)d
Step 2: Find the 7th term (T7)
T7 = a + (7-1)d
Step 3: Write an equation for the sum of T6 and T7
T6 + T7 = 60
Step 4: Substitute the expressions for T6 and T7 into the equation
(a + (6-1)d) + (a + (7-1)d) = 60
Step 5: Simplify the equation
2a + 12d = 60
Step 6: Rearrange the equation to isolate the common difference (d)
12d = 60 - 2a
d = (60 - 2a) / 12
Step 7: Substitute the value of the 3rd term into the equation
-5 = a + (3-1)d
Step 8: Simplify the equation
-5 = a + 2d
Step 9: Rearrange the equation to isolate the first term (a)
a = -5 - 2d
Step 10: Substitute the expression for a into the equation for d
d = (60 - 2(-5 - 2d)) / 12
Step 11: Simplify and solve for d
d = (60 + 10 + 4d) / 12
12d = 70 + 4d
8d = 70
d = 70 / 8
d = 8.75
Therefore, the common difference in the arithmetic progression is 8.75.
To find the common difference of an arithmetic progression, we need to use the information given.
Let's start by understanding what an arithmetic progression is. An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant.
Let's assume the first term of the arithmetic progression is "a" and the common difference is "d".
Given:
The 3rd term of the arithmetic progression is -5. Therefore, we can say the 3rd term is a + 2d because the common difference applies twice to reach the 3rd term.
So, we have:
a + 2d = -5 --(1)
Also:
The sum of the 6th and 7th terms of the arithmetic progression is 60. The 6th term is a + 5d, and the 7th term is a + 6d.
So, we have:
(a + 5d) + (a + 6d) = 60
2a + 11d = 60 --(2)
Now we have a system of equations with two variables (a and d). We can solve these equations simultaneously to find the values of a and d.
Let's solve the system of equations (1) and (2):
Multiply equation (1) by 2 and subtract equation (2) from it to eliminate "a":
2(a + 2d) - (2a + 11d) = -10 - 60
2a + 4d - 2a - 11d = -70
-7d = -70
Divide both sides by -7:
d = -70 / -7 = 10
Therefore, the common difference (d) of the arithmetic progression is 10.
"the sum of the 6th and 7th terms of an arithmetic progression is 60" --> a+5d + a+6d = 60
2a + 11d = 60 , #1
" the 3rd term is -5 " ---> a+2d = -5 , #2
Solve the two equations in two unknowns using your usual method.
I would double the 2nd and subtract them