Question

The sum of the ages of 'n' siblings of a family is equal to 140 years. If the ages of these 'n' siblings are integers that form an arithmetic progression with a common
difference of years, which of the following is a valid pair of values of 'n' and 'd'?​

a. 4,7
b.7,7,
c. 10,2
d. 14,1​

Answer 1

If a is the first term, n is the no. of terms and d is the common difference of an AP, then the sum of first n terms is given by,

$\longrightarrow S_n=\dfrac{n}{2}\left[2a+(n-1)d\right]$

or,

$\longrightarrow S_n=n\left[\dfrac{2a+(n-1)d}{2}\right]$

$\longrightarrow S_n=n\left[a+\left(\dfrac{n-1}{2}\right)d\right]$

In the question the sum is given 140.

$\longrightarrow 140=n\left[a+\left(\dfrac{n-1}{2}\right)d\right]$

We put each value for n and d as in the options to check if a is a positive integer (since a is an age).

Take $n=4$ and $d=7.$ Then,

$\longrightarrow140=4\left[a+\dfrac{3}{2}\cdot7\right]$

$\longrightarrow a+\dfrac{21}{2}=35$

Here a is not a positive integer. Hence option (a) is wrong.

Take $n=7$ and $d=7.$ Then,

$\longrightarrow140=7\left[a+\dfrac{3}{2}\cdot7\right]$

$\longrightarrow a+\dfrac{21}{2}=20$

Here a is not a positive integer. Hence option (b) is wrong.

Take $n=10$ and $d=2.$ Then,

$\longrightarrow140=10\left[a+\dfrac{3}{2}\cdot2\right]$

$\longrightarrow a+3=14$

Here a is a positive integer. Hence (c) is correct.

Take $n=14$ and $d=1.$ Then,

$\longrightarrow140=14\left[a+\dfrac{3}{2}\cdot1\right]$

$\longrightarrow a+\dfrac{3}{2}=10$

Here a is not a positive integer. Hence (d) is wrong.

Hence (c) is the correct answer.

Answer 2

Given : The sum of the ages of 'n' siblings of a family is equal to 140 years. If the ages of these 'n' siblings are integers that form an arithmetic progression with a common difference of years,

To Find : which of the following is a valid pair of values of 'n' and 'd'?​

a. 4,7

b.7,7,

c. 10,2

d. 14,1​

Solution:

Let say ages are

a , a + d , a + d , _________ a + (n-1)d

Sum = (n/2)(2a  + (n-1)d)  =  140

=> n (2a + (n - 1)d) = 280

check  n = 4 and  d = 7

=> 4(2a + 3*7)  = 280

=> 2a  + 21  = 70

=> 2a = 49

=> a = 24.5

not an integer

check  n = 7 and  d = 7

7(2a + 6*7)  = 280

=> 2a  + 42 = 40

=> 2a = -2

=> a = -1

Age can  not be negative

check  n = 10 and  d = 2

10(2a + 9*2)  = 280

=> 2a  + 18 = 28

=> 2a = 10

=> a = 5

Satisfies

check  n = 14 and  d = 1

14(2a + 13*1)  = 280

=> 2a  + 13 = 20

=> 2a = 7

=> a = 2.5

not an integer

Hence valid pair of values of 'n' and 'd' is 10 , 2

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