Question

2n² + 3n is the sum of n terms of an arithmetic sequence. Find common difference​

Answer 1

Answer:

If S(n) = 2n² + 3n then, find the common difference (d) of the AP.

Answer:

d = 4

Note:

• A sequence in which, the difference between the consecutive terms are same is called AP (Arithmetic Progression).

• Any AP is given as ; a , (a + d) , (a + 2d) , .....

• The nth term of an AP is given by ;

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

• The common difference of an AP is given by ;

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

• The sum of first n terms of an AP is given by ;

S(n) = (n/2)[2a + (n-1)d] .

• The nth term of an AP is given by ;

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

Solution:

It is given that ;

S(n) = 2n² + 3n

Thus;

S(n-1) = 2(n-1)² + 3(n-1)

Also;

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

=> T(n) = [2n² + 3n] - [2(n-1)² + 3(n-1)]

=> T(n) = 2n² + 3n - 2(n-1)² - 3(n-1)

=> T(n) = 2[n² - (n-1)²] + 3[n - (n-1)]

=> T(n) = 2[n+(n-1)]•[n-(n-1)] + 3[n - (n-1)]

=> T(n) = 2(n+n-1)(n-n+1) + 3(n-n+1)

=> T(n) = 2(2n-1) + 3

=> T(n) = 4n - 4 + 3

=> T(n) = 4n - 1

Thus,

T(n-1) = 4(n-1) - 1

Now;

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

=> d = [4n - 1] - [4(n-1) - 1]

=> d = 4n - 1 - 4(n-1) + 1

=> d = 4n - 1 - 4n + 4 + 1

=> d = 4

Hence,

The common difference of the AP is 4 , ie ;

d = 4 .

Step-by-step explanation:

Answer 2

Answer:

dont know sorry

Step-by-step explanation:

pls mark brainliest