Help with airthmetic progression query

Question

I am currently working on solving questions on Arithmetic Progression and need to understand where I am going wrong with below derivation.

Let us say there are two A.P.

• AP1 => First term a, common difference d
• AP2 => First term A, common difference D

If first 5 terms of AP1 are a-2d, a-d, a, a+d, a+2d and first 5 terms of AP2 are A-2D, A-D, A, A+D, A+2D

• Sum of 5 terms of AP1 = 5a
• Sum of 5 terms of AP2 = 5A

Then is it correct to say that ratio of first 5 terms of any two APs in the world would always be related to ratio of their first terms.

But when I apply the same on two APs as shown below, above derivation fails.

• AP1 => 2,4,6,8,10 (i.e sum = 30)
• AP2 => 7,10,13,16,19 (i.e. sum = 65)

so while ratio of sum is 30/65, but ratio of first two terms is 2/7 which are not related

I am sure I am making a basic mistake but not able to understand where I am going wrong.

Appreciate your help on this.

Thanks

Answer 1

You have defined $a$ and $A$ as the third, not first, terms of the progressions. The ratio of the sums of the first five terms is equal to the ratio of the third terms (assuming the denominator is not zero). The ratio $\frac {30}{65}$ of the sums is equal to the ratio $\frac {6}{13}$ of the third terms.