Question

6. Find the sum of n term of the AP.5,2,-1, 4-7​

Answer 1

Given A.P is

5,2,−1,−4,−7,...…..

number of terms of A.P is n

first term of this A.P is a

1

=5

second term of this A.P is a

2

=2

common difference

d=a

2

−a

1

=2−5=−3

we know that sum of n term of an A.P is given by

S

n

=

2

n

[2a+(n−1)d)]

⟹S

n

=

2

n

[2×5+(n−1)×−3)]

⟹S

n

=

2

n

[10−3n+3)]

hence the sum of n terms of given A.P is .

⟹S

n

=

2

n

[13−3n]

Answer 2

Step-by-step explanation:

 Given:−       

Radii of two circles = 48cm and 13cm

\underline{\sf \ \ \ \star\ To\ Find :- \ \ \ \ \ \ \ }   ⋆ To Find:−       

We have to find out the Area of that circle whose circumference is equal to the Difference of the circumference of given two circles

\underline{\sf \ \ \ \star\ Solution :- \ \ \ \ \ \ \ }   ⋆ Solution:−       

Find the Circumference of the two circles

\underline{\boxed{\sf{\dag\ \ Circumference\ of \ circle = 2 \pi r}}}†  Circumference of circle=2πr

Find the circumference of circle whose radius is 48cm

\begin{gathered}\dashrightarrow\sf Circumference\ of \ Circle_1= 2\times \dfrac{22}{7}\times 48\\ \\ \\ \dashrightarrow\sf Circumference \ of \ Circle_1={\underline{\boxed{\purple{\sf \dfrac{2112}{7}}}}}\end{gathered}⇢Circumference of Circle1=2×722×48⇢Circumference of Circle1=72112

Find the circumference of circle whose radius is 13cm

\begin{gathered}\dashrightarrow\sf Circumference\ of \ Circle_2= 2\times \dfrac{22}{7}\times 13\\ \\ \\ \dashrightarrow\sf Circumference \ of \ Circle_2={\underline{\boxed{\red{\sf \dfrac{572}{7}}}}}\end{gathered}⇢Circumference of Circle2=2×722×13⇢Circumference of Circle2=7572

Now find out the circumference of new circle which is equal to the Difference of \sf C_1 - C_2C1−C2

\underline{\sf{\maltese \ Circumference\ of \ new \ circle= C_1- C_2}}✠ Circumference of new circle=C1−C2

\begin{gathered}:\implies\sf Circumference\ of \ new \ circle = \bigg[ \dfrac{2112}{7}\bigg]- \bigg[\dfrac{572}{7}\bigg]\\ \\ \\ :\implies\sf C.\ of \ new \ circle = \cancel{\dfrac{1540}{7}}\\ \\ \\ :\implies\sf C.\ of \ new \ circle = {\underline{\boxed{\purple{\sf 220cm}}}}\end{gathered}:⟹Circumference of new circle=[72112]−[7572]:⟹C. of new circle=71540:⟹C. of new circle=220cm

$$\rule{300}{1.5}$$

Now we have to find the Area of new circle

Find out the radius !

$$\begin{gathered}\dashrightarrow\sf Circumference\ of \ circle= 2 \pi r\\ \\ \\ \dashrightarrow\sf 220= 2\times \dfrac{22}{7}\times r \\ \\ \\\dashrightarrow\sf r= \dfrac{\cancel{220}\times 7}{\cancel{44}}\\ \\ \\ \dashrightarrow\sf r= 5\times 7\\ \\ \\\dashrightarrow{\underline{\boxed{\sf{\blue{ radius= 35cm}}}}}\end{gathered}$$

Now find the Area of new circle

$$\underline{\boxed{\sf{\ Area\ of \ circle= \pi r^2 }}}$$

$$\begin{gathered}\dashrightarrow\sf Area \ of \ circle= \dfrac{22}{\cancel{7}}\times \cancel{35}\times 35\\ \\ \\ \dashrightarrow\sf Area\ of \ circle = 22\times 5\times 35\\ \\ \\ \dashrightarrow\sf Area_{circle}= {\underline{\boxed{\sf{\purple{3850 cm^2}}}}}\end{gathered}$$

$$\underline{\underline{\textsf{ Area \ of \ new \ circle = {\textbf{3850sq.cm}}}}}$$

$$\rule{300}{1}$$

$$\underline{\sf{\bigstar\ Alternate\ Method \ To \ find \ Radius \ of \ new \ circle }}$$

$$\begin{gathered}\dashrightarrow\sf C_1- C_2= C_{new}\\ \\ \\ \dashrightarrow\sf 2\pi r_1-2\pi r_2= 2\pi r\\ \\ \\ \dashrightarrow\sf 2\pi(r_1-r_2)= 2\pi r\\ \\ \\ \dashrightarrow\sf \cancel{2 \pi}(48-13)= \cancel{2 \pi } r\ \ \ \ \Big[\therefore\ r_1=48\ ; \ r_2= 13 \Big]\\ \\ \\ \dashrightarrow{\boxed{\sf 35= r}}\end{gathered}$$

★By using this You can easily find the area of the new circle !