The third term of an A.P is 10times more than the second term. Find the sum of the eight and fifteen term of the A.P if the seventh term is seven times the first term
If the second term of an arithmetic sequence is -2 and the fourth term is 6, find the seventh term.
a+2d = 10(a+d)
a+6d = 7a
Solve for a and d, and then you can find the sum of
T8+T15 = (a+7d)+(a+14d)
The third term of an ap is 18 and seveententh term is 30 find the sum of 17 term
To solve this problem, we need to break it down into several steps:
Step 1: Understand the problem
The problem is asking us to find the sum of the eighth and fifteenth terms of an arithmetic progression (A.P.).
Step 2: Define the variables
Let's define the second term of the A.P. as "a," and the common difference as "d."
Step 3: Establish relationships between terms
From the given information, we can determine the relationship between the terms:
The third term is 10 times more than the second term, which can be expressed as:
a + 10d = 10a
The seventh term is 7 times the first term, which can be expressed as:
a + 6d = 7a
Step 4: Solve the equations
We now have a system of two equations with two variables. Let's solve it:
From the first equation, we can simplify it to:
9a = 10d
a = (10/9)d
Substitute this value of "a" into the second equation:
(10/9)d + 6d = 7((10/9)d)
10d + 54d/9 = 70d/9
Multiply through by 9 to remove the fraction:
90d + 54d = 70d
144d = 70d
144d - 70d = 0
74d = 0
Therefore, d = 0.
Step 5: Find the terms
Since the common difference (d) is 0, all subsequent terms will be equal to the second term. Hence, the eighth term and fifteenth term will also be equal to the second term.
Step 6: Calculate the sum of the terms
Since all three terms are the same, the sum of the eight and fifteenth term will be equal to twice the second term:
2a
Step 7: Finalize the solution
To find the value of "a," we can substitute the value of "d" into one of the original equations:
a + 6d = 7a
a + 6(0) = 7a
a = a
This tells us that the value of "a" can be any real number. Therefore, the sum of the eighth and fifteenth terms of the A.P. is dependent on the value of the second term (a), which can vary.