MATHEMATICS : ARITHMETIC PROGRESSIONS

6.The first and the last term of an AP are 17 and 350 respectively. If the
common difference is 9, how many terms are there and what is their
sum?
• Given that,
First term, a = 17
Last term, l = 350
Common difference, d = 9
• Let there be n terms in the A.P., thus the formula for last term can be written as;
l = a+(n −1)d
350 = 17+(n −1)9
333 = (n−1)9
(n−1) = 37
n = 38
Sn = n/2 (a+l)
S38 = 13/2 (17+350)
= 19×367
= 6973
Thus, this A.P. contains 38 terms and the sum of the terms of this A.P. is 6973.

7. Find the sum of first 22 terms of an AP in which d = 7 and
22nd term is 149.
• Common difference, d = 7
22nd term, a22 = 149
Sum of first 22 term, S22 = ?
By the formula of nth term,
an = a+(n−1)d
a22 = a+(22−1)d
149 = a+21×7
149 = a+147
a = 2 = First term
Sum of n terms,
Sn = n/2(a+an)
= 22/2 (2+149)
= 11×151
= 1661

8. Find the sum of first 51 terms of an AP whose second and third terms
are 14 and 18 respectively.
• Given that,
Second term, a2 = 14
Third term, a3 = 18
Common difference, d = a3−a2 = 18−14 = 4
a2 = a+d
14 = a+4
a = 10 = First term
Sum of n terms;
Sn =
𝑛
2
[2a + (n – 1)d]
S51 =
51
2
[2×10 (51-1) 4]
= 51/2[2+(20)×4]
= 51×220/2
= 51×110
= 5610

9. If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289,
find the sum of first n terms
• Given that,
S7 = 49
S17 = 289
We know, Sum of nth term;
Sn = n/2 [2a + (n – 1)d]
Therefore,
S7= 7/2 [2a +(n -1)d]
S7 = 7/2 [2a + (7 -1)d]
49 = 7/2 [2a +16d]
7 = (a+3d)
a + 3d = 7 ………………………. (i)
• In the same way,
S17 = 17/2 [2a+(17-1)d]
289 = 17/2 (2a +16d)
17 = (a+8d)
a +8d = 17 ………………………. (ii)
Subtracting equation (i) from equation (ii),
5d = 10
d = 2
From equation (i), we can write it as;
a+3(2) = 7
a+ 6 = 7
a = 1
Hence,
Sn = n/2[2a+(n-1)d]
= n/2[2(1)+(n – 1)×2]
= n/2(2+2n-2)
= n/2(2n)
= n2

10. 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 the first 15 terms in each case.
• an = 3+4n
a1 = 3+4(1) = 7
a2 = 3+4(2) = 3+8 = 11
a3 = 3+4(3) = 3+12 = 15
a4 = 3+4(4) = 3+16 = 19
• The common difference
between the terms are;
• a2 − a1 = 11−7 = 4
a3 − a2 = 15−11 = 4
a4 − a3 = 19−15 = 4
• Hence, ak + 1 − ak is the same
value every time
AP with common difference as 4 and first term as
7.
Now, we know, the sum of nth term is;
Sn = n/2[2a+(n -1)d]
S15 = 15/2[2(7)+(15-1)×4]
= 15/2[(14)+56]
= 15/2(70)
= 15×35
= 525

(ii) an = 9-5n
• (ii) an = 9−5n
a1 = 9−5×1 = 9−5 = 4
a2 = 9−5×2 = 9−10 = −1
a3 = 9−5×3 = 9−15 = −6
a4 = 9−5×4 = 9−20 = −11
The common difference between the terms are;
• a2 − a1 = −1−4 = −5
a3 − a2 = −6−(−1) = −5
a4 − a3 = −11−(−6) = −5
• Hence, ak + 1 − ak is same every time. An A.P.
with common difference as −5 and first term as
4.
The sum of nth term is;
Sn = n/2 [2a +(n-1)d]
S15 = 15/2[2(4) +(15 -1)(-5)]
= 15/2[8 +14(-5)]
= 15/2(8-70)
= 15/2(-62)
= 15(-31)
= -465

11. If the sum of the first n terms of an AP is 4n − n2, what is the first term (that is S1)?
What is the sum of first two terms? What is the second term? Similarly find the 3rd,
the10th and the nth terms.
• Given that,
Sn = 4n−n2
First term, a = S1 = 4(1) − (1)2 = 4−1 = 3
Sum of first two terms = S2= 4(2)−(2)2 = 8−4 = 4
Second term, a2 = S2 − S1 = 4−3 = 1
Common difference, d = a2−a = 1−3 = −2
Nth term, an = a+(n−1)d
= 3+(n −1)(−2)
= 3−2n +2
= 5−2n
• Therefore, a3 = 5−2(3) = 5-6 = −1
a10 = 5−2(10) = 5−20 = −15
• the sum of first two terms is 4.The second term is 1.
The 3rd, the 10th, and the nth terms are −1, −15, and 5 − 2n respectively

14. Find the sum of the odd numbers between 0 and 50.
.
The odd numbers are in the form of A.P.
First term, a = 1
Common difference, d = 2
Last term, l = 49
By the formula of last term, we know,
l = a+(n−1) d
49 = 1+(n−1)2
48 = 2(n − 1)
n − 1 = 24
n = 25 = Number of terms
sum of nth term, we know,
Sn = n/2(a +l)
S25 = 25/2 (1+49)
= 25(50)/2
=(25)(25)
= 625

If Radha save some money every month in her piggy bank, then how much
money will be there in her piggy bank after 12 months, if the money is in the
sequence of 100, 150, 200, 250, ….respectively?

What is the value of the sum in the nth bracket of (1) + (2+3+4) + (5+6+7+8+9)
+…?

Its Quiz time
• The list of numbers – 10, – 6, – 2, 2, ……….. is an AP with d = _______
• Progressions with equal common difference are known as ________
• The common difference of the A.P. can be _____________
• The first four sequence of 𝑎 𝑛 = 2n + 3 are ________
• The next two terms of the AP : k , 2k +1 , 3k + 2, 4k + 3, ……. are ________
• If a , 7 , b , 23 , c are in AP then the value of ‘c’ is ____

Recall
• Define an Arithmetic Progression?
• What is common difference?
• How to represent a first term?
• How to represent last term?
• How to find the nth term of an AP?
• What is the formula to calculate sum of the nth term of an AP?

Additional Information
A Geometric Progression is a sequence of numbers in which we get each term by
multiplying or dividing a particular number to the previous term, except the first term.
Harmonic Progression It is the reverse of Arithmetic Progression. If a, a + d, a + 2d …..is an
Arithmetic Progression then the harmonic progression is

MATHEMATICS : ARITHMETIC PROGRESSIONS

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