Question 1: Find the sum of the following arithmetic progressions:
Answer:
Question 2: Find the sum of the following series:
Answer:
Question 3: Find the sum of first natural numbers.
Answer:
Question 4: Find the sum of all natural numbers between , which are divisible by
or
Answer:
Sequence of natural numbers between , which are divisible by
is
Sequence of natural numbers between , which are divisible by
is
Sequence of natural numbers between , which are divisible by
is
Hence, sum of all natural numbers between , which are divisible by
or
Question 5: Find the sum of first odd natural numbers.
Answer:
The series would be
Question 6: Find the sum of all odd numbers between
Answer:
Question 7: Show that the sum of all odd integers between which are divisible by
is
Answer:
Question 8: Find the sum of all integers between , which are multiples of
Answer:
Question 9: Find the sum of all integers between which are divisible by
Answer:
Question 10: Find the sum of all even integers between
Answer:
Question 11: Find the sum of all integers between , which are divisible by
Answer:
Question 12: Find the sum of the series:
Answer:
Given sequence
This can we written as
Sum
Question 13: Find the sum of all those integers between each of which on division leaves the remainder
Answer:
The number would be of the form
Question 14: Solve:
Answer:
Let the number of terms be
Let the number of terms be
or
Question 15: Find the term of an A.P., the sum of whose first
is
Answer:
Let the first term and the common difference
Question 16: How many terms are there in the A.P. whose first and fifth terms are respective and the sum of the terms is
?
Answer:
Let the first term and the common difference
or
(this is not possible)
Therefore there are in the A.P.
Question 17: The sum of first of an A.P. is
and that of next
is
Find the progression.
Answer:
… … … … … i)
… … … … … ii)
Solving i) and ii)
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:
… … … … … i)
… … … … … ii)
Solving i) and ii) we get
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:
… … … … … i)
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
Therefore there are in the series.
Now
Therefore series is
Question 21: If and,
, in an A.P., prove that
Answer:
… … … … … i)
… … … … … 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
?
Answer:
… … … … … i)
… … … … … ii)
Solving i) and ii) we get
Question 23: It the of an A.P. are
respectively, what is the sum of the first
?
Answer:
… … … … … i)
Similarly,
… … … … … ii)
Solving i) and ii) we get
Question 24: Find the sum of of the A.P. whose
is
Answer:
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
Question 26: If the sum of a certain number of terms of the A.P. is
Find the last term.
Answer:
Question 27: Find the sum of odd integers from
Answer:
The series will be
Question 28: How many terms of the A.P. are needed to give the sum
Answer:
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:
… … … … … i)
Similarly, … … … … … ii)
Substituting in i)
Question 30: If be the sum of
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:
series is
Question 32: If the sum of n terms of an A.P. is , where
are constants, find the common difference.
Answer:
Question 33: The sums of of two arithmetic progressions are in the ratio
Find the ratio of their
Answer:
Let for first A.P.
Let for second A.P.
Question 34: The sums of first of two A.P.’s are in the ratio
Find the ratio of their
Answer:
Let for first A.P.
Let for second A.P.