Find the Sum of n Terms from the Given Sum of Series ?
From the sum of n terms of the series, we may form a equation and solve for the variables "a" and "d".
To find the sum of n terms of an arithmetic progression, we may use one of the formulas given below.
Sn = (n/2) [a + l] (or)
Sn = (n/2) [2a + (n - 1)d]
a = first term, d = common difference and n = number of terms.
Question 1 :
If the sum of 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.
Solution :
Given that :
S7 = 49
Sn = (n/2) [2a + (n - 1)d]
S7 = (7/2)[2a + (7-1)d]
(7/2)[2a + 6d] = 49
2a + 6d = 49⋅(2/7)
2a + 6d = 14
Dividing both sides by 2
a + 3d = 7 ------(1)
S17 = 289
S17 = (17/2)[2a + (17-1)d]
(17/2)[2a + 16d] = 289
2a + 16d = 289⋅(2/17)
2a + 16d = 34
Dividing both sides by 2
a + 8d = 17 ------(2)
(1) - (2)
a + 3d = 7
a + 8d = 17
(-) (-) (-)
------------------
-5d = -10
d = 2
By applying the value of d in (1), we get
a + 3(2) = 7
a + 6 = 7
a = 7 - 6 = 1
By applying the values of "a" and "d" in the formula for sum of n terms.
Sn = (n/2) [2a + (n - 1)d]
Sn = (n/2) [2(1) + (n - 1)2]
Sn = (n/2) [2 + 2n - 2]
Sn = n2
Question 2 :
Show that a1, a2,............ an form an AP where an is defined as below
(i) an = 3 + 4n
(ii) an = 9 - 5n
Also find the sum of 15 terms in each case.
Solution :
(i) an = 3 + 4n
From the given general term, we may find the first term and common difference.
By applying n = 1, we get the first term.
n = 1 a1 = 3 + 4(1) = 7 |
n = 15 a2 = 3 + 4(15) = 63 |
First term (a) = 7, Last term (l) = 63
Number of terms = 15
Sn = (n/2) [a + l]
S15 = (15/2) [7 + 63]
= (15/2) (70)
= 15(35)
= 525
Hence the sum of 15 terms is 525.
(ii) an = 9 - 5n
From the given general term, we may find the first term and common difference.
By applying n = 1, we get the first term.
n = 1 a1 = 9 - 5(1) = 4 |
n = 15 a2 = 9 - 5(15) = = 9 - 75 = -66 |
First term (a) = 4,
Last term (l) = -66
Number of terms = 15
Sn = (n/2) [a + l]
S15 = (15/2) [4 - 66]
= (15/2) [-62]
= 15 (-31)
= -465
Hence the sum of 15 terms is -465.
After having gone through the stuff given above, we hope that the students would have understood, find the sum of n terms from the given sum of series.
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