Question

The 8th term of ap is equal to three time its third term if its 6th term is 22 find AP

Answer 1

SOLUTION:-

Let a be the first term & d be the common difference of the A.P.

Given:

${}^{a} 8 = 3 {}^{a} 3$

We know that, formula of the Arithmetic Progression;

$= > {}^{a} n = a + (n - 1)d$

=) a + (8-1)d = 3(a+ 2d)

=) a + 7d = 3a + 6d

=) a -3a = 6d -7d

=) -2a = -d

=) 2a = d.............(1)

&

Given:

${}^{a} 6 = 22$

=) a + 5d = 22............(2)

From equation (1) & (2), we get;

$= > \frac{d}{2} + 5d = 22 \\ \\ = > \frac{d + 10d}{2} = 22 \\ \\ = > d + 10d = 44 \\ \\ = > 11d = 44 \\ \\ = > d = \frac{44}{11} \\ \\ = > d = 4$

Putting the value of d in equation (1), we get;

=) 2a = 4

=) a= 4/2

=) a= 2

So,

${}^{a} 2 = a + d \\ \\ = > 2 + 4 = 6 \\ \\ {}^{a} 3 = a + 2d \\ \\ = > 2 + 2(4) \\ \\ = > 2 + 8 = 10$

Thus,

The A.P. is 2, 6, 10......

Hope it helps ☺️

Answer 2

Step-by-step explanation:

Let a be the first term & d be the common difference of the A.P.

Given:

{}^{a} 8 = 3 {}^{a} 3

a

8=3

a

3

We know that, formula of the Arithmetic Progression;

= > {}^{a} n = a + (n - 1)d=>

a

n=a+(n−1)d

=) a + (8-1)d = 3(a+ 2d)

=) a + 7d = 3a + 6d

=) a -3a = 6d -7d

=) -2a = -d

=) 2a = d.............(1)

&

Given:

{}^{a} 6 = 22

a

6=22

=) a + 5d = 22............(2)

From equation (1) & (2), we get;

\begin{gathered} = > \frac{d}{2} + 5d = 22 \\ \\ = > \frac{d + 10d}{2} = 22 \\ \\ = > d + 10d = 44 \\ \\ = > 11d = 44 \\ \\ = > d = \frac{44}{11} \\ \\ = > d = 4\end{gathered}

=>

2

d

+5d=22

=>

2

d+10d

=22

=>d+10d=44

=>11d=44

=>d=

11

44

=>d=4

Putting the value of d in equation (1), we get;

=) 2a = 4

=) a= 4/2

=) a= 2

So,

\begin{gathered} {}^{a} 2 = a + d \\ \\ = > 2 + 4 = 6 \\ \\ {}^{a} 3 = a + 2d \\ \\ = > 2 + 2(4) \\ \\ = > 2 + 8 = 10\end{gathered}

a

2=a+d

=>2+4=6

a

3=a+2d

=>2+2(4)

=>2+8=10

Thus,

The A.P. is 2, 6, 10......

Hope it helps ☺️