in an AP, if the 5th and 12th terms are 30 and 65 respectively. what is the sum of first 20terms
An arithmetic progression (A.P) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.
If the initial term of an arithmetic progression is a1 and the common difference of successive members is d, then the nth term of the sequence is given by:
a n = a 1 + ( n - 1 ) d
a 1 = first term
In this case:
a 5 = 30
a 1 + ( 5 - 1 ) d = 30
a 1 + 4 d = 30
a 1 = 30 - 4 d
a 12 = 65
a 1 + ( 12 - 1 ) d = 65
a 1 + 11 d = 65
30 - 4 d + 11 d = 65
30 + 7 d = 65
7 d = 65 - 30
7 d = 35 Divide both sides by 7
d = 35 / 7
d = 5
a 1 = 30 - 4 d
a 1 = 30 - 4 * 5
a 1 = 30 - 20
a 1 = 10
The sum S of the first n values of a finite sequence is given by the formula:
S n = ( n / 2 ) * [ 2 a 1 + ( n - 1 ) d ]
In tis case :
a 1 = 10
d = 5
n = 20
S 20 = ( 20 / 2 ) * [ 2 * 10 + ( 20 - 1 ) 5 ]
S 20 = 10 * ( 20 + 19 * 5 )
S 20 = 10 * ( 20 + 95 )
S 20 = 10 * 115
S 20 = 1150
To find the sum of the first 20 terms of an arithmetic progression (AP), you need to know the first term (a) and the common difference (d).
In this problem, you are given two specific terms: the 5th term (a₅ = 30) and the 12th term (a₁₂ = 65).
To find the common difference (d), you can use the formula:
d = (a₁₂ - a₅) / (12 - 5)
= (65 - 30) / 7
= 35 / 7
= 5
Now that you have the common difference (d = 5), you can find the first term (a). To do this, you can use the formula:
a = a₅ - 4d
= 30 - 4 * 5
= 30 - 20
= 10
Now you have the first term (a = 10) and the common difference (d = 5) of the given AP.
To find the sum of the first 20 terms, you can use the formula for the sum of an AP:
Sₙ = (n/2) * (2a + (n-1)d)
Here, n = 20 (the number of terms), a = 10 (the first term), and d = 5 (the common difference).
Sₙ = (20/2) * (2 * 10 + (20-1) * 5)
= 10 * (20 + 19 * 5)
= 10 * (20 + 95)
= 10 * 115
= 1150
Therefore, the sum of the first 20 terms of the given AP is 1150.
To find the sum of the first 20 terms in an arithmetic progression (AP), we need two pieces of information: the first term and the common difference. Given that the 5th term is 30 and the 12th term is 65, we can find these values.
Step 1: Finding the common difference (d)
Since the 5th term (a5) is 30 and the 12th term (a12) is 65, we can use the formula for the nth term of an AP: an = a1 + (n-1)d.
Substituting the values, we get:
a5 = a1 + (5-1)d
30 = a1 + 4d ...(1)
a12 = a1 + (12-1)d
65 = a1 + 11d ...(2)
Step 2: Solving for the common difference (d)
Subtracting equation (1) from equation (2), we get:
65 - 30 = (a1 + 11d) - (a1 + 4d)
35 = 11d - 4d
35 = 7d
Simplifying the equation, we find:
d = 35/7
d = 5
Step 3: Finding the first term (a1)
Substituting the value of the common difference (d) into equation (1), we can solve for a1:
30 = a1 + 4(5)
30 = a1 + 20
a1 = 10
Step 4: Finding the sum of the first 20 terms (S20)
The sum of the first 20 terms can be found using the formula:
Sn = n/2[2a1 + (n-1)d]
Substituting the values, we get:
S20 = 20/2 [2(10) + (20-1)(5)]
S20 = 10 [20 + 19(5)]
S20 = 10 [20 + 95]
S20 = 10 [115]
S20 = 1150
Therefore, the sum of the first 20 terms in the AP is 1150.