Question 1: Find the sum of the following arithmetic progressions:
i) to
terms
ii) to
terms
iii) to
terms
iv) to
terms
v) to
terms
vi) to
terms
vii) to
terms
Answer:
i) Given series
Given
We know
ii) Given series
Given
We know
iii) Given series
Given
We know
iv) Given series
Given
We know
v) Given series
Given
We know
vi) Given series
Given
We know
vii) Given series
Given
We know
Question 2: Find the sum of the following series:
i)
ii)
iii)
Answer:
i) Given series
Therefore
We know
We know
ii) Given series
Therefore
We know
We know
iii) Given series
Therefore
We know
We know
Question 3: Find the sum of first natural numbers.
Answer:
Given series
Therefore
We know
Question 4: Find the sum of all natural numbers between and
, which are divisible by
or
.
Answer:
Sequence of natural numbers between and
, which are divisible by
is
Therefore
We know
We know
Sequence of natural numbers between and
, which are divisible by
is
Therefore
We know
We know
Sequence of natural numbers between and
, which are divisible by
is
Therefore
We know
We know
Hence, sum of all natural numbers between and
, which are divisible by
or
Question 5: Find the sum of first odd natural numbers.
Answer:
The series would be
Therefore
We know
Question 6: Find the sum of all odd numbers between and
.
Answer:
The series is
Therefore
We know
We know
Question 7: Show that the sum of all odd integers between and
which are divisible by
is
.
Answer:
The series is
Therefore
We know
We know
Question 8: Find the sum of all integers between and
, which are multiples of
.
Answer:
The series is
Therefore
We know
We know
Question 9: Find the sum of all integers between and
which are divisible by
.
Answer:
The series is
Therefore
We know
We know
Question 10: Find the sum of all even integers between and
.
Answer:
The series is
Therefore
We know
We know
Question 11: Find the sum of all integers between and
, which are divisible by
.
Answer:
The series is
Therefore
We know
We know
Question 12: Find the sum of the series: to
terms.
Answer:
Given sequence
This can we written as
For
For
For
Sum
Question 13: Find the sum of all those integers between and
each of which on division leaves the remainder
Answer:
The number would be of the form
The series is
Therefore
We know
We know
Question 14: Solve: (i) (ii)
.
Answer:
(i)
Let the number of terms be
We know
Since
We know
(ii)
Let the number of terms be
We know
or
Since
We know
Question 15: Find the term of an A.P., the sum of whose first
terms is
.
Answer:
Let the first term and the common difference
Given
Question 16: How many terms are there in the A.P. whose first and fifth terms are and
respective and the sum of the terms is
?
Answer:
Let the first term and the common difference
Given
We know
or
(this is not possible)
Therefore there are terms in the A.P.
Question 17: The sum of first terms of an A.P. is
and that of next
terms is
. Find the progression.
Answer:
Given
… … … … … i)
Also
… … … … … ii)
Solving i) and ii)
Therefore the series is
Question 18: The third term of an A .P. is and the seventh term exceeds three times the third term by
. Find the first term, the common difference and the sum of first
term.
Answer:
Given
We know
… … … … … i)
Also
… … … … … ii)
Solving i) and ii) we get
We know
and
Question 19: The first term of an A.P. is and the last term is
. The sum of all these terms is
. Find the common difference.
Answer:
Given
… … … … … i)
We know
Therefore common difference is .
Question 20: The number of terms of an A.P. is even; the sum of odd terms is , of the even terms is
, and the last term exceeds the first by
, find the number of term and the series.
Answer:
Let the total number of terms be
Let … … … … … i)
… … … … … ii)
Subtracting i) from ii) we get
Also
Therefore there are terms in the series.
Now
Therefore series is
Question 21: If and,
, in an A.P., prove that
.
Answer:
Given
… … … … … i)
Also
… … … … … ii)
Solving i) and ii) we get
… … … … … iii)
Substituting back in i) we get
Question 22: If term of an A.P. is
and the sum of the first four terms is
, what is the sum of first
terms ?
Answer:
Given and
We know
… … … … … i)
We know
… … … … … ii)
Solving i) and ii) we get
Question 23: It the and
terms of an A.P. are
and
respectively, what is the sum of the first
terms ?
Answer:
Given and
We know
… … … … … i)
Similarly,
… … … … … ii)
Solving i) and ii) we get
Question 24: Find the sum of terms of the A.P. whose
terms is
.
Answer:
Given
For
For
Question 25: Find the sum of all two digit numbers which when divided by , yields
as remainder.
Answer:
The number would be of the form
Therefore the series would be for a two digit number
We know
We know
Question 26: If the sum of a certain number of terms of the A.P. is
. Find the last term.
Answer:
Given series
We know
Since
Question 27: Find the sum of odd integers from to
.
Answer:
The series will be
We know
We know
Question 28: How many terms of the A.P.
are needed to give the sum
.
Answer:
Given series
We know
or
Question 29: In an A.P. the first term is and the sum of the first five terms is one fourth of the next five terms. Show that
term is
.
Answer:
Given
We know
… … … … … i)
Similarly,
… … … … … ii)
Substituting in i)
We know
Question 30: If be the sum of
terms of an A.P. and
be the sum of its odd terms, then prove that:
.
Answer:
Let the A.P. be
… … … … … i)
Similarly,
… … … … … ii)
. Hence proved.
Question 31: Find an A.P. in which the sum of any number of terms is always three times the squared number of these terms.
Answer:
Given
For
For
For
series is
Question 32: If the sum of n terms of an A.P. is
, where
and
are constants, find the common difference.
Answer:
Given
For
For
Question 33: The sums of terms of two arithmetic progressions are in the ratio
. Find the ratio of their
terms.
Answer:
Let for first A.P. and
Let for second A.P. and
Given
If you put we get
Question 34: The sums of first terms of two A.P.’s are in the ratio
. Find the ratio of their
terms.
Answer:
Let for first A.P. and
Let for second A.P. and
Given
If you put we get