Question

Sum of first 30 and 40 terms are 2265 and 4020 find the common difference

Answer 1

$\bold{AP}$

Ap is a abbreviation of arithmetic progression.It is a series which same difference between next term and perivous term is always constant.

$\bold{IN \: AN \: AP}$

Sum of n terms = $\boxed{ \frac{n}{2}(2a + (n-1)d}$

ACCORDING TO QUESTION,

sum of first 30 terms = 2265

$\mathsf{2265 = \dfrac{30}{2}(2a + (30-1)d}$

$\mathsf{2265 = 15 \times (2a + 29d)}$

$\mathsf{\dfrac{2265}{15} = 2a + 29d}$

$\mathsf{151= 2a + 29d}$------(1)

sum of first 40 terms = 4020

$\mathsf{4050 = \dfrac{40}{2}(2a + (40-1)d}$

$\mathsf{4020 = 20 \times (2a + 39d)}$

$\mathsf{\dfrac{4020}{20} = 2a + 39d}$

$\mathsf{201= 2a + 39d}$----(2)

On subtracting eq(2) by eq(1)

$\mathsf{ 201= 2a + 39d}$

$\mathsf{151= 2a + 29d}$

__________________________

50 = 10d

___________________________

$\mathsf{\dfrac{50}{10} = d}$

$\boxed{d = 5}$

So, the common difference is 5.

For a ,

$\mathsf{ 201= 2a + 39 \times 5 }$

$\mathsf{ 201= 2a + 195}$

$\mathsf{ 201-195= 2a }$

$\mathsf{ 6 = 2a }$

$\boxed{a = 2}$

$\bold{AP \: will \: be }$

a = 2

a+ d = 2+5 = 7

a + 2d = 2 + 10 = 12

a + 3d = 2 + 15 = 17

$\boxed{2 , 7 , 12 ,17 .......}$