Arithmetic Progression

The digits of a three-digit number are in arithmetic progression. If you divide the number by the sum of its digits, the quotient is 26. If the digits are reversed, the resulting number is 198 more than the original number. Find the sum of all the digits.

A.   9 C.   15
B.   12 D.   18


There are 7 arithmetic means between 3 and 35. What is the sum of all the terms?

A.   133 C.   665
B.   608 D.   171


How many terms from the progression 3, 5, 7, 9, ... must be taken in order that their sum will be 2600?

A.   80 C.   50
B.   60 D.   70


Numbers 4, 2, 5, and 18 are Added Respectively to the First Four Terms of AP, Forming Into a GP

If 4, 2, 5, and 18 are added respectively to the first four terms of an arithmetic progression, the resulting series is a geometric progression. What is the common difference of the arithmetic progression?

A job posted at offered a starting salary of \$40,000 per year and guaranteeing a raise of \$1600 per year for the rest of 5 years. Write the general term for the arithmetic sequence that models potential annual salaries.

A.   an = 38,400 + 1600n
B.   an = 33,400 + 2600n
C.   an = 36,400 + 1400n
D.   an = 34,400 +1800n

Three-digit numbers not divisible by 3

How many three-digit numbers are not divisible by 3?

Arithmetic, geometric, and harmonic progressions

a1 = value of the first term
am = value of any term after the first term but before the last term
an = value of the last term
n = total number of terms
m = mth term after the first but before nth
d = common difference of arithmetic progression
r = common ratio of geometric progression
S = sum of the 1st n terms

Relationship Between Arithmetic Mean, Harmonic Mean, and Geometric Mean of Two Numbers

For two numbers x and y, let x, a, y be a sequence of three numbers. If x, a, y is an arithmetic progression then 'a' is called arithmetic mean. If x, a, y is a geometric progression then 'a' is called geometric mean. If x, a, y form a harmonic progression then 'a' is called harmonic mean.

Let AM = arithmetic mean, GM = geometric mean, and HM = harmonic mean. The relationship between the three is given by the formula

$AM \times HM = GM^2$


Below is the derivation of this relationship.

Derivation of Sum of Arithmetic Progression

Arithmetic Progression, AP

Arithmetic Progression (also called arithmetic sequence), is a sequence of numbers such that the difference between any two consecutive terms is constant. Each term therefore in an arithmetic progression will increase or decrease at a constant value called the common difference, d.

Examples of arithmetic progression are:

  • 2, 5, 8, 11,... common difference = 3
  • 23, 19, 15, 11,... common difference = -4
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