10th Grade > Mathematics
ARITHMETIC PROGRESSIONS MCQs
:
B
The series can be clubbed into two AP's, (1 + 7 + 13......) and (4 + 5 + 6.......). So, we can observe that the series can be taken as two AP's by simple rearrangement of numbers. Now, we can find the Sum of 50 terms for each AP formed.
Formula for sum of first n terms of AP is: Sn=n2(2a+(n−1)d)
So, for first AP
S1=502(2+49×6)=7400
and similarly,
S2=502(8+49)=1425
So, Sum of first 100 terms
=7400+1425=8825
:
B
34+32+30+....+10
This sequence is an AP.
Here, a=34,d=32−34=−2.
Let the number of terms of the AP be n.
We know that
an=a+(n−1)d.
⇒10=34+(n−1)(−2)
⇒(n−1)(−2)=−24
⇒n−1=242=12
⇒n=13
Also, we know that the sum to n terms of an AP with first term a and last term l is given by
Sn=n2(a+l).
∴S13=132(34+10)
S13=286
Hence, the required sum is 286.
:
B
Let S10 be the sum of first 10 terms and S5 be the sum of first 5 terms.
The sum upto n terms of an AP of first term a and common difference d is given by
Sn=n2[2a+(n−1)d].
Given, S10=4S5.
⇒102[2a+(10−1)d]=4×52[2a+(5−1)d]
⇒102[2a+9d]=4×52[2a+4d]
⇒ 2a+9d=4a+8d
⇒ 2a=d
⇒ ad=12
:
D
The nth term of an A.P, with first term a and common difference d is given by
Tn=a+(n−1)d.
⇒a11=a+10d=35...(1)
a13=a+12d=41...(2)
Solving equations (1) and (2), we get
2d=6.
⇒d=3
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It is also given that a3=12 and a50=106
Using formula an=a+(n−1)d to find nth term of AP, we get
a50=a+(50−1)d⇒106=a+49d⋯(1)
a3=a+(3−1)d⇒12=a+2d⋯(2)
Subtracting (1) and (2), we get,
⇒47d=94
⇒d=2
Substituting d in (2), we get,
a+2(2)=12
a=12−4=8
The 29th term is,
a29=a+(29−1)d=8+28(2)=8+56=64
:
D
There are five terms in this AP
an = a1 + (n - 1)d
⇒ an−a1=(n−1)d
⇒ an−a1n−1=d
Here, a1=1; n=5; an=9
Therefore the common difference,
d=9−15−1=2
:
It is given that 17th term exceeds its 10th term by 7
It means a17 = a10 + 7......(1)
Here, a17 is the 17th term and a10 is the 10th term of an AP.
Using formula an=a+(n−1)d to find nth term of arithmetic progression, we get
a17 =a+(16)d .....(2)
a10 = a+(9)d .....(3)
Putting (2) and (3) in equation (1), we get
a+16d=a+9d+7
⇒7d=7
⇒d=77=1