Question

if the sumof first 10 termd of an A.Pis 175 &the sum of next ten terms is 475. Find the A.P​

Answer 1

Answer:

A.P. is 4, 7, 10, 13, .......

Step-by-step explanation:

We are given that sum of first 10 terms of an A.P is 175 & the sum of next ten terms is 475 i.e. ;  $S_1_0$ = 175  and  $S_1_1_-_2_0$ = 475

Here 11 - 20 means sum of terms from 11th to 20th.

Now, Sum of first 20 terms = Sum of first 10 terms + Sum of next ten terms

                                    $S_2_0$    =  $S_1_0$  +  $S_1_1_-_2_0$

                                    $S_2_0$   =  175  +  475 = 650

Now, Sum of n terms of an A.P. formula is given by, $S_n$ = $\frac{n}{2}[2a + (n-1) d]$

So,   $S_1_0$ = $\frac{10}{2}[2a + (10-1) d]$

        175 = $5[2a + 9 d]$

         2*a + 9*d = 35 ------------ [Equation 1]

Also, $S_2_0$ = $\frac{20}{2}[2a + (20-1) d]$

         650 = $10[2a + 19 d]$

         2*a + 19*d = 65 ------------ [Equation 2]    

Solving both the equations we get,

           35 - 9*d = 65 - 19*d

              10*d = 30

                  d = 3  

Putting d in equation 1 we get, a = $\frac{35-9*3}{2}$ = 4

A.P. is given by a, a + d, a + 2*d, a + 3*d,..........

SO, A.P. is 4, 7, 10, 13, ...............

Answer 2

Solution :-

Sum of first 10 terms of an AP = S₁₀ = 175

Using Sum of n terms of an AP formula

Sₙ = n/2(2a+ (n - 1)d)

⇒ S₁₀ = 10/2(2a + (10 - 1)d)

⇒ 175 = 5(2a + 9d)

⇒ 175/5 = 2a + 9d

⇒ 2a + 9d = 35 --- eq(1)

Sum of next 10 terms = 475

We know that

Sum of first 20 terms S₂₀= Sum of first 10 terms + Sum of next 10 terms = 175 + 475 = 650

Using Sum of n terms of an AP formula

Sₙ = n/2(2a+ (n - 1)d)

⇒ S₂₀ = 20/2(2a + (20 - 1)d)

⇒ 650 = 10(2a + 19d)

⇒ 2a + 19d = 65 ---- eq(2)

Subtracting eq(1) from eq(2)

⇒ 2a + 19d - 2a - 9d = 65 - 35

⇒ 10d = 30

⇒ d = 3

Substituting d = 3 in eq(1)

⇒ 2a + 9d = 35

⇒ 2a + 9(3) = 35

⇒ 2a + 27 = 35

⇒ 2a = 35 - 27

⇒ 2a = 8

⇒ a = 8/2 = 4

General form of AP : a, a + d, a + 2d, a + 3d, a + 4d,....

Therefore, the AP is 4, 7, 10, 13, 16, ......