Question 2: Sum of the first 3 terms of an AP is \(\frac{2}{9}\) times the sum of first 6 terms of the same AP. Find the ratio of the first term to the common difference of the same AP.
Answer (2)
The ratio of the first term to the common difference of the AP is 5 1 . This was derived from the sum of the first three and six terms of the arithmetic progression. We used the relationship between the terms to solve for the ratio.
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To solve this problem, we need to find the ratio between the first term a and the common difference d of an arithmetic progression (AP).
In an AP, the sum of the first n terms S n is given by:
S n = 2 n ( 2 a + ( n − 1 ) d )
Let's start by writing the sum for the first 3 terms S 3 and the sum for the first 6 terms S 6 :
Sum of the first 3 terms : S 3 = 2 3 ( 2 a + 2 d ) = 3 ( a + d )
Sum of the first 6 terms : S 6 = 2 6 ( 2 a + 5 d ) = 3 ( 2 a + 5 d )
According to the problem, the sum of the first 3 terms is 9 2 times the sum of the first 6 terms:
3 ( a + d ) = 9 2 × 3 ( 2 a + 5 d )
We can simplify this equation:
Multiply both sides by 9 to eliminate the fraction: 27 ( a + d ) = 2 × 3 ( 2 a + 5 d )
Simplify it further: 27 a + 27 d = 6 ( 2 a + 5 d )
Expand the right side: 27 a + 27 d = 12 a + 30 d
Now, set the terms with a on one side and d on the other:
Subtract 12 a from both sides: 15 a + 27 d = 30 d
Subtract 27 d from both sides: 15 a = 3 d
Divide both sides by 3 to simplify: 5 a = d
Thus, the ratio of the first term a to the common difference d is:
d a = 5 1
This means that the first term a is one-fifth of the common difference d .
