Questions For Practise

# Formulas
1.     Sum of 1st n natural numbers Sn = 1+2+3+……. to n terms =   
2.     Sum of 1st n even numbers Sn = 1 + 2 + 4 + to n terms = .
3.     Sum of 1st n odd numbers Sn = 1 + 3 + 5 +…… to n terms = n2
4.     Sum of square 1st n natural numbers Sn= 12 + 22 + 32 +…………to n terms =
5.     Sum of cubes of 1st n natural numbers Sn = 13 + 23 + 33 +………to n terms= [ ] 2
6.       =
7.      2 =
8.      3 = [ ] 2

# Rules to find General term
To find general terms we use following rules:
1)     If the series/sequence is AS then tn= a + (n-1) d
2)     If the series/sequence is GS then tn = arn-1
3)     If the series/sequence is not both AS and GS we find a pattern.
4)    If we cannot use all these methods mentioned above then tn = an2 + bn2 + c.

# Illustrative Examples
Find the sum of first 5 natural numbers.
Solution:
Here n =5
So, 
Find the sum of first 10 even numbers.
Solution:
Here n = 10
So,
What is the sum of first 7 odd numbers?
Solution:
Here n = 7
So,
Calculate the sum of square of first three natural numbers?
Solution:
Here n = 3

Find the sum of cubes of first 5 natural numbers.
Solution:
Here, n = 5
So,  =  = 152 = 225

# Important Questions for SLC Examination
Find the sum of the following series
1 + 3 + 5 + ………….. 25 terms
2 + 4 + 6 + ……….. 21 terms
1 + 2 + 3 + …………. 25 terms
 2 + 4 + 6 + ……… 30 terms
1 + 3 + 5 + ………… 50 terms
Find the sum of the following series
13 + 23 + 33 …………… +103
13 + 23 + 33 + …………… 8 terms
12 + 22 + 32 + …………….. + 102
12 + 22 + 32 + ……….. 12 terms
Find the nth term and sum of n terms of following series.
22 + 42 + 62 + ……………… n terms
 (1×2) + (2×3) + (3×4) + ……….. n terms
 (2×4) + (3×5) + (4×6) + ………….. n terms

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Questions For Practise

# Short Answer Questions
Find the 5th term of the sequence 3,-6, 12,-24, …
Find the 7th term of the series 3,6,12,24,…….
Find the arithmetic mean and geometric mean between the two numbers 6 and 54.
If the first term of a geometric series is 4 and the sum of first two term is 36, find the common ratio.
How many terms are there in the series   ?
In a GP, the first term is 7 and the last term is 448 and sum 889, find the common ratio.
Find the 8th term of the series 5,-10,-15, 20 …
If the 2nd and 4th term of a GP are -10 and 20 respectively then find first term, common ratio and 12th term.
If 2nd term and 5th term of a GP are 4 and 32 respectively. Find 8th term
In a GP if the common ratio is 2 and the 8th term is 384, find the first term
Which term of the series   is  ?
Find the sum of the following series:
  +1+4+…… 7 terms
 8 terms
 10terms
8-12+18-…….. 
How many terms of the series 32+48+72+….. will add up to 665 ?
Evaluate the following




In a GP if S6 = 28 and S3 = 1, find a and r.
The first three terms of a G.P. are 2x, 2x + 3 & 2x + 9. Find x & the value of 5th term.
If 4th term of a G.P. is 54 and 6th term is 24, which term is 7 ?






# Long Answer Questions

If the second and fifth term of a GS is 4 and 32, find the series.
Given series is 9+3+1+………..+ .
How many terms are there in the series ?
Calculate the sum
For two unequal numbers show that AM> GM
If the AM and GM between two positive and unequal numbers are 5 and 4 respectively, what are the numbers ?
Find two numbers whose arithmetic mean is 5 and geometric mean is 4.
If the AM and GM between two positive and unequal numbers are 13 and 5 respectively, what are the numbers ?
If the AM and GM between two positive and unequal numbers are   and 6 respectively, what are the numbers ?
The sum of first four terms is 40 and the sum of first two terms is 4 of a geometric series whose common ratio is positive, find the sum of first eight terms.
If the AM of two unequal positive real numbers a and b (a>b) be twice as great as their GM, show that 
Anisha borrows Rs 4368 which she promises to pay in 6 annual instilments, each installment being treble of the preceding one. Find the first and the last installment.
The product of three numbers in a GP is 216 and the sum of products of the numbers taken in pairs is 156. Find the numbers.
x + 6, x and x - 3 are the first three terms of a geometric series. Find the value of x and its fifth term.
If x, 6, y, 24, P are in GS, find the values of x, y, p given that common ratio is +ve.
If 4th term of a G.P. is 54 and 6th term is 24, which term is 7 ?
Insert 3 geometric means between  and 9.
Insert 4 GM between and 16; also find the sum of the series.
Insert 2 GM between 6 and 48.
Insert 3 GM between 5 and 3125.
The 2nd term of a GP is 6 and its 5th term is 162. Find the sum of 1st 5 terms.
If ,x,y,z are in G.P., find the values of x, y & z.
If 5, a, b, 135 are in GP, find a and b.
If 2, m, n, s,  are in GP, find m, n and s.
The sum of 3 consecutive terms of G.P. is 28 and their product is 512. Find the terms
The sum of three terms in AP is 30. If 5 is added to the third term it becomes a GP. Find the terms.
Find two numbers whose AM is 34 & GM is 16.
If a, b, c are in AP and x, y, z are in G.P. prove that.
In a G.P. tp = a, tq = b & tr = c. Then show that aq-r br-p cp-q = 1.

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Questions For Practise For Optional Maths

# Short-Answer Questions:

Find the values of 

Sin15o
Tan750
Sin750
Cos75o
Tan105o
Cos150+Cos75


Show that:

Cos400- Sin400 = Sin50
 Cos15o-Sin15o = 
Sin105o+Cos105o = 
Cos500 + Cos700 = Cos100
Tan15o + Cot15o = 4
Sin105o + Cos75o = 
Tan75o +Cot75o = 4
Cos(400+) – Sin(500-) = 0
 = tan100
 = Cot500


Solve the following

If CosA =  and CosB = , find the value of:
Sin (A-B)                   
Cos (A-B)                       
If CosA =  and CosB =  ,find the value of:
Cos (A+B)     
Sin (A-B)

# Long-Answer Questions:

Prove the following identities:






Prove that:

Sin (45o+ ) +Cos (45o+ ) =  Cos
Cot (45o+ ) = 
Sin (A-45o) = 
Tan (45o+A) tan (45o-A) = 1
Sin2 (45o+A) +Sin2 (45o-A) = 1

Solve the following problems.   
If tan A =  and tanB = , show that A+B = 
If tanA =  and tanB = , show that A+B = 
If tanA = , show that A+B = 
If tanA =   show that A+B = 
Prove the following: 

Cos18o-Sin18o =  Sin27o
Cos55o + Cos35o = Cos10o


Prove the following:






Show that:
Tan20o+tan25o+tan20otan25o=1
Tan20o+tan72o+tan88o = tan20o.tan72otan88o
Tan3A-tan2A-tanA = tan3A.tan2A.tanA
If (A+B) = 45o, show that :
(1+tanA) (1+tanB) = 2
(CotA-1) (CotB-1) = 2

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