There are 6 terms in an A.P. 6th term is 3 times the second term and 4th term is 4 times the first term. if the 5th term is 15 find the terms
2nd term = a+d
6th term = a + 5d
"the 6th term is 3 times the second term" -->
a+5d = 3(a+d)
a+5d = 3a + 3d
2d = 2a
a = d
"the 5th term is 15" --->
a+4d = 15
but a = d
5d = 15
d = 3
then a = 3
"the 4th term is 4 times the first term" -->
a+3d = 4a
3d = 4a
d= a
this last piece of information was not needed, but confirms our answer
the terms are
3 6 9 12 15 and 18
Let's assume the first term of the arithmetic progression (A.P.) is "a" and the common difference is "d".
Given information:
- 6th term (a + 5d) = 3 times the second term (a + d)
- 4th term (a + 3d) = 4 times the first term (a)
Using this information, we can create two equations:
Equation 1: a + 5d = 3(a + d)
Equation 2: a + 3d = 4a
From Equation 1, we expand and simplify:
a + 5d = 3a + 3d
2d = 2a
d = a
Substituting this result into Equation 2:
a + 3(a) = 4a
2a = 3(a)
2a = 3a
a = 0
Therefore, the first term is 0.
Using this information, we can find the common difference (d) and other terms in the A.P.
d = a = 0
Now, we can find the other terms:
2nd term: a + d = 0 + 0 = 0
3rd term: a + 2d = 0 + 0 = 0
4th term: a + 3d = 0 + 0 = 0
5th term: a + 4d = 0 + 0 = 0
6th term: a + 5d = 0 + 0 = 0
Therefore, all the terms in the arithmetic progression are 0.
To find the terms of the Arithmetic Progression (A.P.), we need to analyze the given information step by step.
Let's assume that the first term of the A.P. is 'a', and the common difference between the terms is 'd'. Therefore, the second term would be 'a + d', the third term would be 'a + 2d', and so on.
Given:
1) The 4th term is 4 times the first term: 4a = a + 3d
2) The 6th term is 3 times the second term: 3(a + d) = a + 5d
3) The 5th term is 15: a + 4d = 15
Now, we can use these three equations to solve for 'a' and 'd'.
Equation 1: 4a = a + 3d => 3a - 3d = 0 => a - d = 0 (equation after simplification)
Equation 2: 3(a + d) = a + 5d => 3a + 3d = a + 5d => 2a - 2d = 0 (equation after simplification)
By solving equations 1 and 2 simultaneously, we find that 'a' and 'd' are both equal to zero. However, since terms in a non-trivial A.P cannot be zero, it means that the terms in the given A.P do not exist.
Therefore, there are no terms in the A.P satisfying the given conditions.