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Arithmetic Progressions

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Mastering Arithmetic Progression: Formulas, Examples, and Sum of Sequences

Comprehensive Definition, Description, Examples & Rules 

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A sequence of numbers in which the difference between any two consecutive numbers is identical throughout is known as arithmetic progression or AP. This difference between two consecutive numbers is known as the common difference. a very important concept in math with various uses in varied fields, it comes in handy when dealing with algebra, number theory, calculus, etc., in the real world.

The common difference between any two consecutive numbers in an arithmetic progression is denoted by the letter ‘d’. For an arithmetic progression where A is the first term, the progression can be represented as ‘A, A+ d, A+2d, A+3d…’ and so on. The common difference lays a standard ground essential for understanding how the sequence progresses. The common difference not just gives characteristics to an arithmetic progression but also helps one to analyse and predict its typical behaviour.

Finding the nth Term of an Arithmetic sequence

The arithmetic sequence formula for finding the nth term is:

a_n=a_1+(n-1)*d

Here,

  • a_n is the nth value of the arithmetic progression
  • a_1 is the first value of the arithmetic progression
  • n is the position number in the arithmetic progression
  • d is the standard disparity between two successive values

 

Here are step-by-step arithmetic sequence examples of finding the nth term using the common difference:

  • For an arithmetic progression with 5 as the first term and common difference 2, find the 6th term.

Solution: Using the formula a_n=a_1+(n-1)*d,

            a_6= 5+(6-1)*2

            a_6= 5+5*2

            a_6= 5+10

            a_6=15

  • Find the 6th term of an arithmetic progression if the first term is -3 and the common difference is 9.

Solution: Using the formula a_n=a_1+(n-1)*d,

            a_6= -3+(6-1)*9

            a_6= -3+7*9

            a_6= -3+63

            a_6=60

The Arithmetic Formula for the 'nth Term

The general arithmetic formula for finding the nth term (a_n) of an arithmetic sequence is:

a_n=a_1+(n-1)*d

Here,

  • a_n is the nth value of the arithmetic sequence
  • a_1 is the primary term of the arithmetic sequence
  • n is the numeral of places in the arithmetic sequence
  • d is the standard disparity between two concurrent terms

By employing this arithmetic sequence formula, you can easily calculate the nth term of an arithmetic succession by summing up the derivative of the common difference and the place of the term to the term with which the arithmetic succession commences.

Here are a couple of arithmetic sequence examples to illustrate the application of the formula:

  • For an arithmetic succession with -4 as the first term and common difference 5, find the 9th term.

Solution: Using the formula a_n=a_1+(n-1)*d,

            a_9= -4+(9-1)*5

            a_9= -4+8*5

            a_9= -4+40

            a_9=36

  • Find the 13th term of an arithmetic progression if the first term is 9 and the common difference is 7.
  • Solution: Using the formula a_n=a_1+(n-1)*d,

            a_13= 9+(13-1)*7

            a_13= 9+12*7

            a_13= 9+84

           a_13=93

The Sum of Arithmetic Progression

The aggregate of all the entities in an arithmetic progression is called the sum of that arithmetic progression. Per term of an arithmetic progression is acquired by climbing up a constant numeral to every individual term. The sum of an arithmetic progression is a vital notion in math and is used in varied applications like computing whole lengths, deciphering problems with consistent advancements quickly, etc.

To derive the formula for the sum of ‘n’ terms in an AP, you need to consider the sum of the first ‘n’ terms. So, the sum of sequence formula is: 

S_n= a_1+(a_1+d) + (a_1+2d)+…+[a_1+(n-1)d]

Where,

  • S_n is the sum of the first ‘n’ terms
  • a_1 is the first term
  • d is the standard difference between two successive values
  • n is the number of terms

It should be observed from the formula above that each term is formed by adding the multiples of the common difference, ‘d’, to the first term, ‘a_1’. Hence the formula can also be jotted down as follows:

S_n= n*a_1+d[1+2+3+…+(n-1)] 

So the sum for the first ‘n’ natural numbers is calculated by:

[n*(n-1)] / 2

Substituting these values to the equations, we get:

S_n= n*a_1 + d*[n*(n-1)] / 2

Or, 

S_n= (n/2) * [2*a_1 + (n-1) *d]

The Sum of the Sequence Formula

The sum of the sequence formula is given below:

Sum= (n/2) * (first term of the sequence+ last term of the sequence)

Here,

  • The sum is the total of an arithmetic sequence
  • N is the number of terms in the sequence 

When we compare the sum of an arithmetic sequence with the sum of a general sequence, we find that:

  • The common difference between consecutive terms in an arithmetics sequence remains the same, whereas the formula for a general sequence depends on the nature of the sequence and can be computed through procedures like finding a common factor, the highest common multiple, etc.
  • The sum of an arithmetic sequence depends on the first term, the common difference between consecutive terms and the number of terms, whereas the sum of a general sequence depends on the particular pattern or formula used to create the terms.

Arithmetic sequence Sum Formula

An arithmetic sequence is a set of numbers where every term is attained by adding the same value or common difference to the last term. To find the sum of an arithmetic sequence, follow the steps given below:

  • Count the number of terms in the sequence
  • Note the first number of the sequence
  • Note the last number of the sequence

Now, the sum can be calculated as, Sum= (number of terms / 2) * (first term + last term)

To understand the arithmetic sequence sum formula better, you can imagine an arithmetic sequence on a number line, with every term of the line equally spaced and the first term of the sequence labelled as ‘a’ and the common difference as ‘d’. Or you can also use a slider to control the number of terms ‘n’ in the sequence. Go a step further by visualising how the terms are added, and the sequence grows as you move the slider.

FAQs and common mistakes to watch out for when using the formula:

  • Why should one divide the number of terms by 2?

You need to divide the number of terms by 2 as it stands for counting every term twice as it appears two times, as the first term and the last term. Make sure you divide the number of terms by 2 always to avoid miscounting terms and overestimating the sum.

  • What if the common difference is negative?

You can still use the formula for finding the sum of an arithmetic sequence even if the common difference is negative, as the formula works for any value of ‘d’, be it positive, negative or zero.

nth Term of Arithmetic sequence

The value of the arithmetic sequence at a particular position or term number ‘n’ is what we know as the nth term of that arithmetic sequence. The nth term lets us predict any item in the arithmetic sequence without the need to list all the items in the sequence, which comes in handy in various real-world scenarios.

Here is a step-by-step approach to calculating the nth term:

  • Note the first term of the arithmetic sequence
  • Find out the common difference
  • Calculate with the help of the formula: a_n=a_1+(n-1)*d

 Some examples are:

  • For an arithmetic progression with 5 as the first term and common difference 3, find the 12th term.

Solution: Using the formula a_n=a_1+(n-1)*d,

            a_12= 5+(12-1)*3

            a_12= 5+11*3

            a_12= 5+33

            a_12=38

  • Find the 52nd term of an arithmetic progression if the first term is 13 and the common difference is 22.

Solution: Using the formula a_n=a_1+(n-1)*d,

            a_52= 13+(52-1)*22

            a_52= 13+51*22

            a_52= 13+1122

              a_52=1135

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Key Takeaways

  1. Arithmetic progression is a sequence of numbers in which the difference between two terms is always the same.

  2. The common difference between two consecutive terms is denoted by ‘d’ and the number of terms by ‘n’

  3. Arithmetic sequences have many real-life applications in fields like finance, physics, computer science, etc.

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Frequently Asked Questions

A sequence is an arithmetic progression if the difference between consecutive terms remains the same.

By subtracting any term from its last term, you can find the common difference in an arithmetic sequence.

In an arithmetic progression, terms increase by a constant difference whereas in a geometric progression, terms multiply by a constant ratio.

The first term in an arithmetic sequence is the initial term of the sequence through which the sequence is generated by adding common differences to consecutive terms.

To find the number of terms in an arithmetic progression, use the formula: last term – first term / common difference + 1

Arithmetic progression is used in finance for calculating interests, in computer science for various algorithms, in physics for kinematics, and in various other fields.

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2D Shapes2cosacosb Formula30-60-90 Formulas3D ShapesAbsolute Value FormulaAcute AngleAcute Angle triangleAdditionAlgebra FormulasAlgebra of MatricesAlgebraic EquationsAlgebraic ExpressionsAngle FormulaAnnulusAnova FormulaAnti-derivative FormulaAntiderivative FormulaApplication of DerivativesApplications of IntegrationArc Length FormulaArccot FormulaArctan FormulaArea Formula for QuadrilateralsArea FormulasArea Of A Sector Of A Circle FormulaArea Of An Octagon FormulaArea Of Isosceles TriangleArea Of ShapesArea Under the Curve FormulaArea of RectangleArea of Regular Polygon FormulaArea of TriangleArea of a Circle FormulaArea of a Pentagon FormulaArea of a Square FormulaArea of a Trapezoid FormulaArithmetic Mean FormulaArithmetic ProgressionsArithmetic Sequence Recursive FormulaArithmetic and Geometric ProgressionAscending OrderAssociative Property FormulaAsymptote FormulaAverage Deviation FormulaAverage Rate of Change FormulaAveragesAxioms Of ProbabilityAxis of Symmetry FormulaBasic Math FormulasBasics Of AlgebraBinary FormulaBinomial Probability FormulaBinomial Theorem FormulaBinomial distributionBodmas RuleBoolean AlgebraBusiness MathematicsCalculusCelsius FormulaCentral Angle of a Circle FormulaCentral Limit Theorem FormulaCentroid of a Trapezoid FormulaChain RuleChain Rule FormulaChange of Base FormulaChi Square FormulaCirclesCircumference FormulaCoefficient of Determination FormulaCoefficient of Variation FormulaCofactor FormulaComplete the square formulaComplex numbersCompound Interest FormulaConditional Probability FormulaConeConfidence Interval FormulaCongruence of TrianglesCorrelation Coefficient FormulaCos Double Angle FormulaCos Square theta FormulaCos Theta FormulaCosec Cot FormulaCosecant FormulaCosine FormulaCovariance FormulaCubeCurated Maths Resources for Teachers – EdulyteCylinderDecimalsDifferential calculusDiscover the world of MathsEllipseEquilateral triangleEuler’s formulaEven numbersExponentsFractionFraction to decimalGeometric sequenceHeptagonHyperbolaIntegersIntegrationIntegration by partsLinesLocusMatricesNatural numbersNumber lineOdd numbersParallelogramPercentage formulaPerimeterPolygonPolynomialsPrismProbabilityPyramidPythagoras theoremRoman NumeralsScalene triangleSetsShapes NamesSimple interest formulaSlope formulaSolid shapesSphereSquareStandard deviation formulaSubtractionSymmetryTimeTrianglesTrigonometry formulaTypes of anglesValue of PiVariance formulaVectorVolume formulasVolume of a coneVolume of sphere formulaWhole numbers
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