Question

Write the solutions for the following:
i. Find the sum of the first 8 multiples of 3.
ii. Find the ratio in which P (4, m) divides the line segment joining the points A (2, 3) and B (6, -3). Then find m.

Answer 1

We solve the questions individually, as follows

(i) Find the sum of the first 8 multiples of 3.
Hint: Let's find out the first 8 multiples of 3 or calculate the 8th term by using the first term which will be in the form of A.P with common difference (d) and first term being a. Then we can apply the direct formula of Sum of n terms of an A.P i.e, $\dfrac{{\text{n}}}{2}\left( {{\text{a + }}{{\text{a}}_{\text{n}}}} \right)$

Complete step-by-step answer:

The multiples of 3 are in arithmetic progression with common difference (d) =3, i.e. the difference between any two consecutive terms in the progression is 3.

Now, First multiple of the progression = 3

We know, the nth term of AP is given as = a + (n-1) d, where a is the first term, d is the common difference and n is the total number of terms in the Arithmetic progression.

Hence, 8th multiple of the progression is
= 3 + (8-1) 3
= 3 +21

Hence 8th term = 24

We also know, Sum of first n terms in AP is given as =$\dfrac{{\text{n}}}{2}\left( {{\text{a + }}{{\text{a}}_{\text{n}}}} \right)$, where n is the number of terms in the progression, a is the first term and ${{\text{a}}_{\text{n}}}$is the last term of the progression.

Sum of first 8 multiples of 3
= $\dfrac{8}{2}\left( {3 + 24} \right)$
= 4 x 27
= 108

Hence, Sum of the first 8 multiples of 3 equals to 108.

Note: There is an alternate formula to find sum of n terms of an A.P.
Sum of n terms of AP = n/2[2a + (n – 1)d], where ‘a’ , ‘n’ and ‘d’ being the same as mentioned for earlier formulas in the solution.

(ii) Find the ratio in which P (4, m) divides the line segment joining the points A (2, 3) and B (6, -3). Then find m.

Hint: To find m, we use the section formula to determine the ratio in which P divides the line segment. Then we use the ratio and apply section formula again to determine the y coordinate, i.e. m.

Complete step-by-step answer:

We know section formula states,

Let k: 1 is the ratio by which P divides the line joining A and B, then
x = $\dfrac{{\left( {{\text{k}}{{\text{x}}_2} + {{\text{x}}_1}} \right)}}{{\left( {{\text{k + 1}}} \right)}}$

Applying it to the x coordinate, here, x = 4, ${{\text{x}}_1}$ = 2, ${{\text{x}}_2}$= 6
Now,
$\Rightarrow 4{\text{ = }}\dfrac{{6{\text{k + 2}}}}{{{\text{k + 1}}}} \\\ \Rightarrow 4\left( {{\text{k + 1}}} \right) = 6{\text{k + 2}} \\\ \Rightarrow {\text{4k + 4 = 6k + 2}} \\\ \Rightarrow {\text{2k = 2}} \\\ \Rightarrow {\text{k = 1}} \\\$

Hence, P divides the line AB into 1:1 ratio.

Now, to find y coordinate, i.e. m, we use the same section formula
y =$\dfrac{{\left( {{\text{k}}{{\text{y}}_2} + {{\text{y}}_1}} \right)}}{{\left( {{\text{k + 1}}} \right)}}$, here k = 1, ${{\text{y}}_2} = - 3$ and ${{\text{y}}_1} = 3$

According to the given, y = m
$\Rightarrow {\text{y = }}\dfrac{{\left( {{\text{k}}{{\text{y}}_2} + {{\text{y}}_1}} \right)}}{{\left( {{\text{k + 1}}} \right)}} \\\ \Rightarrow {\text{m = }}\dfrac{{\left( {{\text{1}} \times - 3 + 3} \right)}}{{\left( {{\text{1 + 1}}} \right)}} \\\ \Rightarrow {\text{m = }}\dfrac{0}{2} = 0 \\\$

Hence, m = 0

Note: In order to solve this type of question the key is to identify that we have to apply the section formula. Also, we have to notice the section formula is to be applied separately for each of the co-ordinates. So we consider the ratio to be a variable k, find it by applying the formula on x –coordinate since x is given. Then we use all of this data to find y coordinate i.e. m.