Question

answer my question ​

Answer 1

Step-by-step explanation:

Given -

AP = 3, 15, 27, 39, ........

To Find -

Which term of the AP is 120 more than 53rd term

Now,

As we know that :-

• an = a + (n - 1)d

here,

a = 3

n = 53

d = 15-3 = 12

» an = 3 + (53 - 1)12

» 3 + 52 × 12

» 3 + 624

• » 627

Hence,

53rd term of the AP is 607

And

We need to find :-

Which term of the AP is 120 more than 53rd term

It means,

» 627 + 120

• » 747

Hence,

747 is 120 more than 53rd term.

Now,

We have to find which term of AP is 747

As we know that :-

• an = a + (n - 1)d

» 747 = 3 + (n - 1)12

» 747 - 3 = 12n - 12

» 744 + 12 = 12n

» 756 = 12n

» n = 756/12

• » n = 63

Hence,

63rd term of the AP will be 120 more than its 53rd term.

Answer 2

Concept & Formula To Remember For AP :-

• A sequence is said to be in AP (Arithmetic Progression), if the difference between its consecutive terms are equal.

• The nth term of an AP is given as ;

T(n) = a + (n-1)•d , where a is the first term and d is the common difference.

• The common difference of an AP is given as ;

d = T(n) - T(n-1)

• If the number of terms in an AP is n ( where n is odd ) ,then there will be a single middle term.

Also, [(n+1)/2]th term will be its middle term.

• If the number of terms in an AP is n ( where n is even ) ,then there will be two middle terms.

Also, (n/2)th and (n/2 + 1)th terms will be its middle terms.

• The sum up to nth terms of an AP is given as ;

S(n) = (n/2)•[2a + (n-1)•d] where a is the first term and d is the common difference.

• The nth term of an AP is also given as ;

T(n) = S(n) - S(n-1)

_________________

Solution :-

Given That :-

⟼ First Term = a₁= 3

⟼ Second Term = a₂ = 15 .

⟼ common Difference = d = ( a₂ - a₁) = 15 - 3 = 12.

➻ T(n) = a + (n-1)•d

➻ T₅₃ = 3 + (53 - 1)12

➻ T₅₃ = 3 + 52*12

➻ T₅₃ = 3 + 624

➻ T₅₃ = 627.

So,

➼ Number 120 More Than T₅₃ = 627 + 120 = 747.

________________

Now, we Have to Find which Term of AP is 747.

Again,

➪ T(n) = a + (n-1)•d

➪ 747 = 3 + (n - 1)12

➪ 747 = 3 + 12n - 12

➪ 747 = 12n - 9

➪ 12n = 747 + 9

➪ 12n = 756

➪ n = 63. (Ans.)

Hence, 63rd Term of AP is 120 More Than 53rd Term.

[ Note :- Common Difference is 12 , and We have to find 120 more than 53rd Term, since in AP common difference b/w Numbers Remains same. So, we can say that , sum of 120 is distributed among (120/12)=10 such Numbers and than our Final Term will be 53rd + 10 = 63rd Term. ]