**Arithmetic progression and geometric progression formulas :**

On the webpage, we can find the formulas used in the topic arithmetic and geometric progression.

**General form of arithmetic progression :**

**a , (a+d), (a+2d), (a+3d), ................**

**nth term or general term of the arithmetic sequence :**

**an = a+(n-1)d**

here "n" stands for the required term.** **

**Number of terms in the arithmetic sequence :**

**n = [(l- a)/d]+1 **

**Common difference :**

**d = a₂ - a₁**** **

**In the above formula,**

**"a" stands for the first term****"d" stands for the common difference**- "l" stands for the last term
- and "n" stands for the total number of terms or required term.
- a₁ and a₂ are first and second term respectively.

Note:

Suppose if we want to find the 15th term of the given sequence, we need to apply n = 15 in the general term formula.

**General form of geometric progression :**

**a , ar, ar**², .............

**Common ratio :**

**r = a₂ / a₁**** **

**nth term or general term of the arithmetic sequence :**

**an = ar^(n-1)**

In the above formula,

**"a" stands for the first term****"r" stands for the common ratio**- "n" stands for the required term
- a₁ and a₂ are first and second term respectively.

**Question 1 :**

Find the common difference and 15th term of an A.P 125 , 120 ,115 , 110 , ……….….

**Solution :**

First term (a) = 125

Common difference (d) = a2 – a1 ==> 120 – 125 ==> -25

General term of an A.P (an) = a + (n - 1) d

= 125 + (15 - 1) (-25) ==> 125 + 14 (-25) ==> 125 – 350

a₁₅ = -225

Therefore 15th term of A.P is -225

**Question 2 :**

Which term of the arithmetic sequence is 24 , 23 ¼ ,22 ½ , 21 ¾ , ………. Is 3?

**Solution :**

First term (a) = 24

Common difference = a2 – a1 ==> 23 ¼ – 24 ==> (93/4) – 24

d = -3/4

an = a + (n - 1) d

Let us consider 3 as nth term

an = 3

3 = 24 + (n-1) (-3/4)

3 – 24 = (n-1) (-3/4)

(-21 x 4)/(-3) = n -1 ==> 84/3 = n -1 ==> 28 = n – 1 ==> n=29

Hence,3 is the 29th term of A.P.

**Question 3 :**

The 10th and 18th terms of an A.P are 41 and 73 respectively. Find the 27th term and ap

**Solution :**

10th term = 41 ==> a + 9 d = 41 ------- (1)

18th term = 73 ==> a + 17 d = 73 ------- (2)

Subtracting the second equation from first equation

aaaaaaaaaaaaaaaaa a + 17d = 73 aaaaaaaaaaaaaaaaaaaaaaaa

aaaaaaaaaa(-)aaaa aa + 9 d = 41 aaaaaaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaa(-)a(-)aaa(-)aaaaaaaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaa------------ aaaaaaaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaa a8d = 32 aaaaaaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaaaaad = 4aaaaaaaaaaaaaaaaaaaaaaaaa

Substitute d = 4 in the first equation

a + 9 (4) = 41 ==> a + 36 = 41 ==> a = 5

Now, we have to find 27th term

an = a + (n - 1) d

here n = 27

= 5 + (27-1) 4 ==> 5 + 26 (4) ==> 5 + 104 ==> 109

Hence, 27th term of the sequence is 109.

General form of ap:

a, (a+d), (a+2d),...............

5, (5+4), (5+8), ...............

5, 9, 13,....................

Let us see the next example on "Arithmetic progression and geometric progression formulas".

**Question 4 :**

Find n so that the nth terms of the following two A.P’s are the same

1 , 7 ,13 ,19, ………………. and 100 , 95 , 90 ,………..

**Solution :**

an = a + (n - 1) d

nth term of the first sequence

a = 1 d = t₂-t₁ ==> 7-1 ==> d = 6

an = 1 + (n-1) 6 ==> 1 + 6 n – 6 ==> 6 n – 5 -----(1)

nth term of the second sequence

a = 100 d = a₂-a₁ ==> 95 - 100 ==> -5

an = 100 + (n-1) (-5) ==> 100 - 5 n + 5 ==> 105 - 5 n -----(2)

(1) = (2)

6 n – 5 = 105 – 5 n

6 n + 5 n = 105 + 5

11 n = 110 ==> 110/11 ==> 11

Hence, 11th terms of the given sequence are equal

Let us see the next example on "Arithmetic progression and geometric progression formulas".

**Question 5 :**

7, 13, 9,............................205

**Solution :**

First term a = 7,

common difference d = t2 - t1 = 13 - 7 = 6

l = 205

n = [(l-a)/d] + 1

n = [(205 - 7)/6] + 1

n = [198/6] + 1 ==> 33 + 1 ==> 34

Hence, total number of terms in the above sequence is 34

**Question 6 :**

Find the 10th term and the common ratio of the geometric sequence 1/4,-1/2,1,-2,............

**Solution :**

To find the 10th terms of the G.P we have to use the formula for general term that is

tn = a r^(n-1)

here a = 1/4 r = (-1/2)/(1/4) ==>(-1/2) x (4/1) = -2

n = 10

t₁₀ = (1/4) (-2)^(10-1)

= (1/4) (-2)^9

= (1/4) (-512)

= -512/4

= -128

**Question 7 : **

If the 4th and 7th terms of a G.P are 54 and 1458 respectively, find the G.P

**Solution :**

** **4th term = 54

7th term = 1458

t₄ = 54

a r³ = 54 ----- (1)

t₇ = 1458

a r⁶ = 1458 ----- (2)

(2)/(1) = (a r⁶)/(a r³) = 1458/54

r³ = 27

r³ = 3³

r = 3

Substitute r = 3 in the first equation we get

a (3)³ = 54

a(27) = 54

a = 54/27

a = 2

The general form of G.P is a, a r , a r ²,.........

= 2 ,2(3),2(3)²,..............

= 2,6,18,............

After having gone through the stuff given above, we hope that the students would have understood "Arithmetic progression and geometric progression formulas".

Apart from the stuff given above, if you want to know more about "Arithmetic progression and geometric progression formulas", please click here

Apart from the stuff "Arithmetic progression and geometric progression formulas" given in this section, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

**WORD PROBLEMS**

**HCF and LCM word problems**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**