Progressions level 2

PROGRESSIONS
LEVEL 2
C.S.VEERARAGAVAN

Nth term of A.P
What is the seventh term of an arithmetic progression
whose first term is 9 and the common difference is 3?
1) 27 2) 36 3) 33 4) 30
Tn = a + ( n – 1) d
T7 = 9 + ( 7 – 1) 3
= 9 + 18
= 27
09
Friday, June 12, 2015
VEERARAGAVAN C S veeraa1729@gmail.com
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A.M of A.P
What is the arithmetic mean of the arithmetic progression
6,8,10,12,14,16? a) 22 b) 11 c) 24 d) 12
A.M =
6+8+10+12+14+16
6
=
66
6
= 11
01
Shortcut : A.Mean of the middle terms
Or
First term+Last term
2
A.M =
10+12
2
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A.M of an A.P
What is the fourth term of the arithmetic progression in
which the first term is 4 and the seventh term is 28?
1) 16 2) 8 3) 12 4) 2
The fourth term is the A.M of the First and Seventh term.
Fourth term =
4+28
2
= 16
03
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A.M of A.P
What is the arithmetic mean of an arithmetic progression
with 13 terms in which the 7th term is 9?
1) 9 2)
91
7
3)
95
7
4) Cannot be determined
Given the total number of items is (n) = 13.
In A.P if n is odd, then the arithmetic mean is
𝑛+1
2
𝑡ℎ
term.
Here it is 7th term.
Hence A.M = 9.
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G.M of G.P
What is the geometric mean of the Geometic Progression
2,4,8,16? 1) 32 2) 32 3) 64 4) 8
G.M = 2∗4∗8∗16
1
4 = 2
10
4 = 32
02
Shortcut : G.Mean of the middle terms
Or First term ∗Last term
G.M = 2 ∗ 16
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Sum of terms of A.P
What is the sum to 7 terms of an ‘arithmetic progression’ in
which the first term is 2 and the common difference is 4?
1) 49 2) 98 3) 126 4) 77
Sn =
n
2
2a + n−1 d
S7 =
7
2
2∗2 + 7−1 4
=
7
2
4+24
= 98
04
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Sum of terms of A.P
What is the sum to 8 terms of an ‘arithmetic progression’ in
which the first term is 3 and the last term is 31?
1) 136 2) 58.5 3)132 4)Cannot be determined
Since number of terms is unknown, we cannot find the
sum.
05
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Sum of terms of A.P
What is the sum of all the terms of an ‘arithmetic
progression’ in which the first term is 5, the last term is 15
and the number of terms is 11?
1) 55 2) 110 3) 115 4) Cannot be determined
Sn =
n
2
First term+Last term
S11 =
11
2
5 + 15
S11 = 110
08
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Sum of terms of A.P
If the sum of first 20 terms of an A.P is 30 and the sum of
first 50 terms is also 30, then what is the sum of 21st term
and the 50th term?
1) 0 2) 30 3) 15 4) Cannot be determined.
Given S20 = 30 and S50 = 30.
S50 = S20 + t21 + t22 + … t50 = 30
= 30 + t21 + t22 + … t50 = 30
= t21 + t22 + … t50 = 0
Mean of t21 and t50 = 0.
Sum of t21 and t50 = 0.
16
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Sum of terms of A.P using A.M
What is the sum to 15 terms of an arithmetic progression
whose 8th term is 4?
The eighth term is equidistant from first and last terms.
So 8th term is the A.M of first 15 terms.
Sum of first 15 terms = 15 x (8th term)
= 15 x 4
= 60.
07
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Sum of terms of A.P using A.M
In an A.P having 100 terms, the mth term from the
beginning and the mth term from the end are 10 and 20
respectively.
What is the sum of all the terms?
Since average of mth term from both ends is A.M
A.M =
10+20
2
= 15.
Sn = n ∗ A.M
= 100 * 15 = 1500
12
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nth term of a G.P using G.M
What is the fourth term of a geometric progression in which
the second term is 4 and the sixth term is 64?
1) 8 2) 32 3) 64 4) 16
Tn = a * r( n – 1).
T2 = ar = 4 & T6 = ar5 = 64.
T6
T2
= r4 = 16.
Hence r = 2 and a = 2
T4 = ar3 = 16
10
Shortcut: Fourth Term is
equidistant from second and
sixth term.
G.M = 4∗64
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Sum of terms of G.P
What is the sum to 4 terms of a Geometric Progression
whose first term is 6 and the common ratio is 3?
1) 300 2) 360 3) 270 4) 240
S4 =
a rn−1
r−1
= 6
3
4
−1
2
= 6 * 40 = 240.
11
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A.P or G.P ?
If m, n and p are in A.P, then the mth term, the nth term and
the pth term of A.P are in
1) A.P 2) G.P 3) not necessarily in A.P or G.P
Given m,n,p are in A.P
n – m = p – n = k.
n = m + k and p = m + 2k.
Let a1, a2, a3, … be A.P and common difference is d.
am+1 = am + d, am+2 = am + 2d…
an = am+k = am + kd and ap = am + 2kd.
an – am = kd and ap – an = kd. Hence they are in A.P
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A.P or G.P?
If m, n and p are in A.P, then the mth term, the nth term and
the pth term of G.P are in
1) A.P 2) G.P 3) not necessarily in A.P or G.P
Given m,n,p are in A.P
n – m = p – n = k.
n = m + k and p = m + 2k.
Let a1, a2, a3, … be G.P and common difference is r.
am+1 = am * r , am+2 = am * r2 …
an = am+k = am * rk and ap = am+2k = am * r2k
Hence they are in G.P
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A.P
If the positive numbers m, n and p are in G.P, then
log m , log n , and log p are in
1) A.P 2) G.P 3) Cannot be determined
Given m, n, p are in G.P
n
m
=
p
n
= r.
Hence n = mr and p = mr2.
Let log m = a and log r = b
Log n = log mr = log m + log r = a + b
Log p = log m + 2 log r = a + 2b
Hence log m, log n, log p are in A.P
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Which term of the A.P is ‘0’.
If the sum of first 51 terms of an A.P is zero, then which of
the following terms is zero?
1) 13th 2) 26th 3) 17th 4) Cannot be determined
In A.P If sum of n terms is zero, and n is odd,
n+1
2
th term is also zero.
Hence 26th term is zero.
15
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G.M and A.M
If the Geometric Mean of two distinct positive numbers is 4,
then the arithmetic mean of these two numbers is
1) < 4 2) = 4 3) > 4 4) Cannot be determined
Always A.M > G.M
𝑎 + 𝑏
2
> 𝑎𝑏
A.M > 4
18
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Nth term of G.P
What is the seventh term of a G.P whose first term is 3 and
the common ratio is 2?
1) 96 2) 384 3) 192 4) 288
a = 3 and r = 2.
T7 = ar6
= 3 * 26
= 3 * 64 = 192.
19
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G.P
What is the product of first 9 terms of a G.P having a total
of 13 terms given that 5th term is 2?
5th term is equidistant from both first and 9th terms.
Hence G.M of first 9 terms is the 5th term = 2.
Product of the nine terms = 29 = 512.
20
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G.M
What is the geometric mean of the Geometric Progression
having a total of 13 terms given the 7th term is 4?
In G.P if the number of terms is odd, then the middle term
is the G.M.
Hence G.M = 7th term = 4.
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G.P
What is the sum of infinite geometric series
1,
1
2
,
1
4
,
1
8
,
1
16
,…?
1) 1
255
266
2) 2 3) 3 4) 4
a = 1 and r =
1
2
S∝ =
a
1−r
=
1
1
2
= 2
22
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G.P
Every number of an infinite geometric progression of
positive terms is equal to m times the sum of the
numbers that follow it.
What is the common ratio of the progression?
1)
m
m+1
2)
1
m+1
3)
2
m+1
4) Cannot be determined
Let the G.P be a , ar, ar2, ar3,…
Given a = m (ar + ar2 + ar3 + …)
ar + ar2 + ar3 + … =
ar
1−r
a =
mar
1−r
 1 – r = mr
1 = (m + 1) r  r =
1
m+1
23
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Sum of the terms of G.P
What is the sum to 7 terms of a G.P whose first term is 1
and the 4th term is 27?
1) 1093 2) 2186 3) 3279 4) 4372
Given a = 1 and ar3 = 27.
Hence r = 3
S7 =
a r7−1
r−1
=
3
7
−1
3−1
=
2187 −1
2
= 1093
24
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Sum of the first n cubes
What is the sum of the cubes of first 9 natural numbers?
Sum of the first 9 natural numbers =
n n+1
2
= 45
Sum of the cubes of first 9 numbers =
n n+1
2
2
= 452
= 2025
25
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