Question

The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms.

Answer 1

Answer:

The Common Difference is  4 , First term is - 1 and Sum of first 20 terms is 740.

Step-by-step explanation:

Given :  

a3 = 7 and a7 =  3(a3) + 2

By using the formula , nth term of an AP is ,an = a + (n – 1)d

Case 1 :

a3 = 7

⇒ a + (3 - 1)d = 7

⇒ a + 2d = 7   

⇒ a = 7 - 2d . …………(1)

Case 2 :

⇒ a7 = 3a3 + 2

⇒ a + (7 - 1)d = 3(7) + 2

⇒ a + 6d = 21 + 2

⇒ a + 6d = 23

⇒ 7 – 2d + 6d = 23

[From eq 1]

⇒ 7 + 4d = 23

⇒ 4d = 23 - 7

⇒ 4d = 16

⇒ d = 16/4

⇒ d = 4

Common Difference ,d = 4

On putting the value of d = 4 in equation 1,

a = 7 - 2d

a = 7 – 2(4)

a = 7 - 8

a = -1

First term ,a = - 1

By using the formula ,Sum of nth terms , Sn = n/2 [2a + (n – 1) d]

Sum of first 20 terms ,S20 = 20/2[2 × -1 + (20 - 1)4

S20 = 10 [-2 + 19 × 4]

S20 = 10[-2 + 76]

S20 = 10 [74]

S20 = 740

Sum of first 20 terms is 740.

Hence, the Common Difference is  4 , First term is - 1 and Sum of first 20 terms is 740.

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Answer 2

Step-by-step explanation:

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