Consider a sequence with first term 7 and common difference 4
(a) write first 5 terms
(b) write 20th term
(c) find sum of first 20 terms
Answers
Answer :
- 7, 11, 15, 19, 23
- 83
- 900
Step-by-step explanation :
We are given that the first term of an arithmetic sequence is 7 and the common difference of two successive terms is 4.
We know that general form of an Arithmetic Progression ( A.P. ) is " an " which is equal to [a + ( n - 1 ) d].
Here,
- a = First term
- d = Common difference
- n = Number of terms
- an = nth term
We will use this general term to solve the given problem.
(a) In the first question, we have to write first 5 terms.
We have a = 7 and d = 4.
According to the general term :
→ an = a + ( n - 1 ) d
Put n = 2 to find 2nd term
→ a2 = 7 + ( 2 - 1 ) 4
→ a2 = 7 + (1) 4
→ a2 = 7 + 4
→ a2 = 11 ----Eqn(1)
Put n = 3 in general term to find 3rd term
→ a3 = 7 + (3 - 1) (4)
→ a3 = 7 + (2) (4)
→ a3 = 7 + 8
→ a4 = 15 ----Eqn(2)
Put n = 4 in general term to find 4th term
→ a4 = 7 + (4 - 1) (4)
→ a4 = 7 + (3) (4)
→ a4 = 7 + 12
→ a4 = 19 ----Eqn(3)
Put n = 5 in general term to find 5th term
→ a5 = 7 + (5 - 1) (4)
→ a5 = 7 + (4) (4)
→ a5 = 7 + 16
→ a5 = 23 ----Eqn(4)
From equations 1, 2, 3 and 4, we get first five terms of AP :
- 7, 11, 15, 19, 23
(b) In this question, we have to write the 20th term of sequence.
We know that nth term of AP is represented as,
→ an = a + (n - 1) d
Substitute n = 20 for finding 20th term
→ a20 = 7 + ( 20 - 1 ) (4)
→ a20 = 7 + (19) (4)
→ a20 = 7 + 76
→ a20 = 83
So the 20th term of given sequence is 83.
(c) Now in this question, we have to find the sum of first 20 terms of AP.
No problem! We have a formula for finding sum also.
Sum of first n terms of an AP is given by :
→ Sn = n/2 [ 2a + ( n - 1 ) d ]
Here,
- Sn = Sum of n terms
- n = number of terms
- a = First term
- d = Common difference
By substituting the given values in the above formula, we get :
→ S20 = 20 / 2 [ 2 ( 7 ) + ( 20 - 1 ) (4 ) ]
→ S20 = 10 [ 14 + (19) (4) ]
→ S20 = 10 [ 14 + 76 ]
→ S20 = 10 [ 90 ]
→ S20 = 900
Hence the sum of 20 terms of given sequence is 900.
Step-by-step explanation:
a) 7,11,18,25,32....
b) 20th term is= a+ 19d
= 7+19(4)
= 7+76
=82
c) the sum of first 20 terms=
sn= n/2[2a+(n-1)d]
=20/2(7×2+19)4
=10(14+76)
=10(90)
=900