Sequence and Series

Sum of the first p terms is q and sum of the first q terms is p of an arithmatic series. 8th term of a geometric series is -27 and 11th term is 81sqrt(3).

  1. Calculate the sum of the first p+q terms of the arithmatic series.
  2. Calculate the sum of the first 9 terms of the geometric series.

Geometric Series

If the ratio of any term to its preceding term is always equal, i.e., if the quotient is always equal when dividing any term by its preceding term, then the series is called a geometric series. The quotient is known as common ratio.

Common ratio = Antecedent term Consequent term = nth term (n-1)th term
If first term of a geometric series is a, and the common ratio is r. Then,
Nth term of the geometric series is: ar^{n-1}
Sum of the first n terms of the geometric series is: a(1-r^n)/(1-r)
Sum up to infinity of a geometric series is: a/1-r; It is calculatable if and only if -1<r<1

Challenge Yourself: Practice Problems to Solidify Concepts of Sequence and Series

  2. A young man saves 1200 taka from his first month's salary and in each of the following months he saves 100 taka more than the previous month.
    1. How much money does he save in the nth month?
    2. Express the problem using a series of n terms.
    3. How much money does he save in the first n number of months?
    4. How much money does he save in a year?
  3. Sum of the first p terms of an arithmatic series is q and the sum of the first q terms is p. Find the sum of the first p+q terms.

    Let,
    First term = a
    Common difference = d
    Sum of the first p terms = p/2(2a+(p-1)d)
    Sum of the first q terms = q/2{2a+(q-1)d}

  4. 8th term of a geometric series is -27 and 11th term is 813. Caluculate sum of the first nine terms of the series.
  5. Find the number of terms and the sum of each of the series given below:
    1. 23 + 27 + 31 + ... + 159
    2. 28 + 11 - 6 - ... - 210
  6. The first two terms in an arithmetic progression are -2 and 5. The last term in the progression is the only number in the progression that is greater than 200. Find the sum of all the terms in the progression.
  7. The first term of an arithmetic progression is 8 and the last term is 34. The sum of the first six terms is 58. Find the number of terms in this progression.