5 times the 5th term of arthmetic progression is equa to 10 times the 10th term. Find the common difference and the first term
5(a+4d) = 10(a+9d)
5a+20d = 10a+90d
a = -14d
So, there is no unique solution. Some examples are
a = 14
d = -1
14,13,12,11,10,9,8,7,6,5
5*10 = 10*5
a = -7
d = 0.5
-7,-6.6,-6,-5.5,-5,-4.5,-4,-3.5,-3.-2.5
5(-5) = 10(-2.5)
To solve this problem, let's assume that the first term of the arithmetic progression is 'a' and the common difference is 'd'.
The formula to find the nth term of an arithmetic progression is given by: an = a + (n-1)d
Given that the 5th term is 5 times the 10th term, we can write the following equation:
5(a + 4d) = 10(a + 9d)
Let's simplify this equation:
5a + 20d = 10a + 90d
Now, let's isolate the variables on one side and the constants on the other side:
20d - 90d = 10a - 5a
-70d = 5a
To find the common difference, we can choose any value for 'a', and then use the relation -70d = 5a to determine the value of 'd'.
Let's assume a value for 'a', say a = 1:
-70d = 5(1)
-70d = 5
d = -5/70
d = -1/14
So, the common difference is -1/14.
To find the first term, substitute the value of 'd' in the relation -70d = 5a:
-70(-1/14) = 5a
10a = 10
a = 1
Therefore, the first term is 1.
In summary, the common difference is -1/14 and the first term is 1.