Question

nth term of Ap is an =8-3n find the common difference ​

Answer 1

EXPLANATION.

Nth terms of an A.P. = 8  - 3n.

As we know that,

Nth terms of an A.P.

Aₙ = a + (n - 1)d.

⇒ 8 - 3n.

Put n = 1 in equation, we get.

⇒ 8 - 3(1).

⇒ 8 - 3.

⇒ 5.

Put n = 2 in equation, we get.

⇒ 8 - 3(2).

⇒ 8 - 6.

⇒ 2.

Put n = 3 in equation, we get.

⇒ 8 - 3(3).

⇒ 8 - 9.

⇒ -1.

Put n = 4 in equation, we get.

⇒ 8 - 3(4).

⇒ 8 - 12.

⇒ -4.

Now, series will be written as.

⇒ 5,2,-1,-4,,,,,,, n terms.

As we know that,

First term = a = 5.

Common difference = d = b - a = c - b.

⇒ 2 - 5 = - 1 - 2.

⇒ -3 = -3.

Common difference = -3.

⇒ Aₙ = a + (n - 1)d.

⇒ aₙ = 5 + (n - 1)(-3).

⇒ aₙ = 5 + [-3n + 3].

⇒ aₙ = 5 - 3n + 3.

⇒ aₙ = 8 - 3n.

HENCE PROVED.

                                                                                                                                             

MORE INFORMATION.

Arithmetic progression (A.P.).

If a is the first term and d is the common difference then A.P can written as,

a + (a + d) + (a + 2d) + (a + 3d) +,,,,,,,

General term of an A.P.

General term (nth term) of an A.P is given by,

Tₙ = a + (n - 1)d.

Sum of n terms of an A.P.

Sₙ = n/2[2a + (n - 1)d]  or  Sₙ = n/2[a + Tₙ].

(1) = If sum of n terms Sₙ is given then general term Tₙ = Sₙ - Sₙ₋₁. where Sₙ₋₁ is sum of (n - 1) terms of an A.P.

Answer 2

Given:-

• $\sf a_{n}$ = 8-3n

To find:-

• Common difference of this A.P

Solution:-

• the 1st term of this A.P will be given by putting 1 in place of n.

$\longrightarrow$ $\sf a_{1}$ = 8-3(1)

$\longrightarrow$ $\sf a_{1}$ = 5

• 2nd term of this A.P will be given by putting 2 in place of n.

$\longrightarrow$ $\sf a_{2}$ = 8-3(2)

$\longrightarrow$ $\sf a_{2}$ = 2

• 3rd term of this A.P will be given by putting 3 in place of n

$\longrightarrow$ $\sf a_{3}$ = 8-3(3)

$\longrightarrow$ $\sf a_{3}$ = -1

hence, the A.P becomes:-

$\longrightarrow$$\underline{\boxed{\sf 5, 2, -1 ... }}$

Now,

$\longrightarrow$ common difference = 2nd term - 1st term

$\longrightarrow$ Common difference = 2-5 = -3

____________________________

Additional information:-

• to find the sum of the terms of an A.P, following formula is used

$\longrightarrow$ $\underline{\boxed{\sf S_{n} = \dfrac{n}{2}×2a+(n-1)d}}$

here,

• $\sf S_{n}$ = Sum of n terms
• n = no. of terms
• a = 1st term
• d = common difference