The 4th term of an Ap is 6. if the sum of the 8th and 9th terms is -72,find the common difference
t(4) = 6 ---- a + 3d = 6
t(8) + t(9) = -72
a+7d + a+8d = -72
2a + 15d = -72
double the first equation
2a + 6d = 12
subtract them
9d = -84
d = -84/9 = -28/3
Please how did you get -28/3
You are right thanks Alot
You are doing well
the first winner lottery get #2000 while subsequently winners get #50 less than the proceeding winner how much will the winner the 20th position get
91/3
To find the common difference of an arithmetic progression (AP), we need to use the given information and apply the formula for the nth term of an AP.
Let's start by writing down what we know:
- The 4th term of the AP is 6.
- The sum of the 8th and 9th terms is -72.
Step 1: Finding the 4th term
We are given that the 4th term of the AP is 6. In an AP, the nth term formula is:
an = a1 + (n - 1)d,
where "an" is the nth term, "a1" is the first term, "n" is the term number, and "d" is the common difference.
Using the given information, we have:
a4 = a1 + (4 - 1)d, (equation 1)
6 = a1 + 3d.
Step 2: Finding the sum of the 8th and 9th terms
We are given that the sum of the 8th and 9th terms is -72. In an AP, the sum of the first "n" terms formula is:
Sn = (n/2) * (2a1 + (n - 1)d),
where "Sn" is the sum of the first "n" terms.
Using the given information, we have:
S8 + S9 = (8/2) * (2a1 + (8 - 1)d) + (9/2) * (2a1 + (9 - 1)d),
-72 = (4) * (2a1 + 7d) + (4.5) * (2a1 + 8d).
Step 3: Solving the equations
Now we have a system of equations with two unknowns: a1 and d.
We can use substitution or elimination to solve these equations.
Let's use elimination method to solve the equations. We need to eliminate either "d" or "a1" from the equations. We can eliminate "a1" by multiplying equation 1 by 4.5 and equation 2 by -3.
Multiplying equation 1 by 4.5 gives:
27 = (4.5a1) + (13.5d). (equation 3)
Multiplying equation 2 by -3 gives:
-216 = -12a1 - 27d. (equation 4)
Next, add equations 3 and 4 to eliminate "a1":
27 + (-216) = (4.5a1 + (-12a1)) + (13.5d + (-27d)),
-189 = -7.5a1 - 13.5d.
Step 4: Finding the common difference
Now we have a single equation containing only the common difference "d":
-189 = -7.5a1 - 13.5d.
Since we don't know the value of "a1", we can focus on finding the common difference "d".
Rearranging the equation, we have:
-13.5d = -189 - (-7.5a1),
-13.5d = -189 + 7.5a1,
13.5d = 7.5a1 - 189,
d = (7.5a1 - 189)/13.5.
So, the common difference of the arithmetic progression is given by the expression (7.5a1 - 189)/13.5, where "a1" is the first term.
Please note that we cannot determine the value of the common difference without knowing the value of the first term "a1" or having additional information.
The first and last terms of an arithmetic progression are 1 and 121 respectively. Find
1.the number of terms in arithmetic progression
2.the common difference between them if d sum of its terms is
(a) 549. (b) 671. (c) 976.
(d) 1281