Question

There are 30 terms in an arithmetic pregression.
The 2nd and 3rd terms are distinct integers.
The ratio of the 1st 20 terms and the sum of the 1st 10 terms equals twice the ratio of the 2nd and the 1st terms.Which of the following cn b the sum of all its terms?

a.1120
b.1560
c.2020
d.3750

Answer 1

is something missing from the third sentence

Answer 2

no mate...dis iz d cmplete question

Answer 3

The ratio of the 1st 20 terms and the sum of the 1st 10 terms equals twice the ratio of the 2nd and the 1st terms

the ratio of what of the 1st 20 terms

Answer 4

20/2(2a+19d)/10/2(2a+9d)=2(a+d)/a

Answer 5

the ratio of the "sums"

Answer 6

ok..give me a min

Answer 7

if you can form equations from above expression you got, you will get a=9d/8. But any idea what's a or d?

Answer 8

I think something is missng

Answer 9

a= first no
nd d= d difference

Answer 10

so 8a=9d or d=0

Answer 11

no mate dis iz d cmplete question @mokeira

Answer 12

ok so you can do this thing, go back to your equation substitute a=9d/8 in place of a and try to get the value of "d" out of it.

Answer 13

i stil wnt gt d ans coz equation wud b 30/2(2a+(30-1)d)=x

Answer 14

from where that x came from? x is what?"

Answer 15

x is the sum of all the 30 terms......i hv to find d sum of all d 30 terms

Answer 16

yeah that won't help..1 min

Answer 17

20/2(2a+19d)/10/2(2a+9d)=2(a+d)/a=d(8a-9d)
d=0 or 8a=9d...dunno hw to proceed after dis

Answer 18

@ganeshie8

Answer 19

d is looking fine to me..

Answer 20

yups dats d ans

Answer 21

Till \(8a = 9d\) you got??

Answer 22

yes

Answer 23

Now, you are given that second and terms are distinct integers..

Note that the words, Distinct Integers, it is an AP so second and third term will be distinct no doubt, but the point to note is, they should be INTEGERS.. and this is the key point here..

Answer 24

*third is missing in first line.. :)

Answer 25

so d is nt equal to 0

Answer 26

The 2nd and 3rd terms are distinct integers.

Answer 27

Your progression will be : a, a+d, a+2d, and so on..

Here, a+d is the second term,
And a + 2d is the third term.. Right??

Answer 28

yes

Answer 29

d is the common difference, it must not be 0 for an AP to form..

Answer 30

I am finding second and third term here..

Answer 31

First term is : a

Answer 32

Similarly can you find now third term??

Answer 33

Wait I have done mistake there..

\[a_2 = a + d = a + \frac{8}{9}a \implies a_2 = \frac{17}{9}a\]

Answer 34

Now find third term here..

Answer 35

Can you find third term here in terms of a?

Answer 36

25/9a

Answer 37

it wud b 25/9a ryt

Answer 38

Yes... Good..

Answer 39

Now you know what Integers are??

Answer 40

There @cj49 ??

Answer 41

yesssss d d third term wud b 25/9a??

Answer 42

Yes I said, you are right..

Answer 43

For second and third term to be Integer, a must be 9 (minimum) or any multiple of 9..

Answer 44

so dat means d shud b an multiple of 8

Answer 45

d is already multiple of 8..

Answer 46

30/2(2a+29d)

Answer 47

See:

\[d = \frac{8a}{9}\]

Factor 8 is there in numerator which shows d is multiple of 8..

Answer 48

lol yes i missed dat

Answer 49

So, a = 9 or 18 or 27 likewise multiples of 9..

So, for least value you can take a = 9..

Answer 50

Just put d = 8 and a = 9 in sum upto 30 terms there..

Answer 51

okkkzzzzzz so D fits d bill...

Answer 52

Note that if you are taking a = 9, then d = 8 there..

Answer 53

tysm :)....

Answer 54

Also your question is pretty much clear that it is asking for what CAN BE the sum and not what WILL BE the sum, that means a and d values will change but a will be multiple of 9 only everytime..

Answer 55

If you have in question given that, what will be the sum, then you will get purely one valid value for a and one for d, but here what can be the sum it is asking, so there can be more values of a and d and hence more values of sum..


Like a = 9 and d = 8 will give d option..
a = 18 and d = 16 will give you some other value.. Likewise.. Hope you are getting what I am trying to say..

Answer 56

its will so i juz hv to substitue d vales nd gt an possible ans nd D is d nli one dat satisfies d cndition

Answer 57

yes i understud dat as d value of a wud change d sum wud also change

Answer 58

Good question ) @cj49

Answer 59

@waterineyes what if common difference is mixed number or suppose it is irrational roots? then how we can suppose minimum value?

Answer 60

We are not concerned with what out d is, you are given that second and third term would be integers and you have those terms as 17a/9 and 25a/9, here to make these terms integers, a must be 9 or multiple of 9, so that denominator 9 will get cancelled and we get a pure integer here..

Now point to note here is, if second term is an integer, suppose let us say second term is 17, can you tell me if d is mixed value, then your third term will definitely going to be mixed which will contradict question's statement.. So, d = integer has to be the case here..

Afterall, if you add an integer to an integer, then only you will get another integer. :)

Answer 61

ok understood )