Course sections

1
The smallest number which when divided by 4, 6 or 7 leaves a remainder of 2, is (1993)
2
Which is the least number that must be subtracted from 1856 so that the remainder, when divided by 7, 12, and 16, is 4? [1994]
3
The remainder obtained when a prime number greater than 6 is divided by 6 is [1995]
4
5^6-1 is divisible by a. 5 b.13 c. 31 d. None of these. breaking a larger exponent (1995)
5
Two positive integers differ by 4 and sum of their reciprocals is 10/21, then one of the numbers is a.  3 b.  1 c.  5 d. 21 [1995]
6
Three consecutive positive even numbers are such that thrice the first number exceeds double the third by 2, then the third number is   [1995].
7
If a number 774958A96B is to be divisible by 8 and 9 (1996)
8
If m and n are integers divisible by 5, which of the following is not necessarily true? a.  m – n is divisible by 5 [1997]
9
Consider a sequence, where the nth term is tn = n/n 2. The value of t3 * t4 * t5 * … t53 equals(2000)
10
A red light flashes 3 times per minute and a green light flashes 5 times in two minutes at regular intervals. If both lights start flashing at the same time, how many times do they flash together in each hour? ( 2001)
11
7^6n- 6^6n, where n is an integer >0, is divisible by (2002)
12
A child was asked to add first few natural numbers (that is, 1 2 3……) so long his patience permitted. As he stopped, he gave the sum as 575. When the teacher declared the result wrong the child discovered he had missed one number in the sequence during addition. The number he missed was: (2002)
13
What is the sum of all two-digit numbers that give a remainder of 3 when they are divided by 7 ? ( 2003)
14
What is the remainder when 4^96 is divided by 6 ? (2003)
15
Which among 2^1/2, 3^1/3, 4^1/4, 12^/12 is the largest? (2004)
16
The remainder, when 15^23 23^23 is divided by 19 is? a.4        b.15        c.0        d.18 (2004)
17
Let S be a set of positive integers such that every element n of S satisfies the conditions (2005)
18
When you reverse the digits of the number 13, the number increases by 18. How many other 2 digit numbers increase by 18 when their digits are reversed?  ( 2006)
19
a/b=1/3, b/c=2, c/d=1/2, d/e=3 then what is the value of abc/def? (2006)
20
Consider four-digit numbers for which the first two digits are equal and the last two digits are also equal. How many are such numbers perfect squares? (2007)
21
how-many-pairs-of-positive-integers-m-n-satisfy 1/m 4/n=1/12, where n is an odd number less than 60…. (2007)
22
What is the number of distinct terms in the expansion of (a b c)^20? (2008)
23
What is the remainder when 7^74 
24
In a tournament, there are n teams T1, T2 ….., Tn with n > 5. Each team consists of k players, k > 3… (2007)
25
a^2-ab b^2/a^2 ab b^2=1/3 then find a/b (CAT, 2009)
26
Let Sn denote the sum of squares of the first n odd natural numbers. If Sn = 533n, find the value of n. (2011)
27
The value of numbers  2^2004  and 5^2004 are written one after another. How many digits are there in all? (2011)
28
If p be a prime number, p>3 and let x be the product of positive number 1, 2, 3,…,(p-1), then consider the following statements (2012)
29
What is the sum of all two digit numbers which leave a remainder of 6 when divided by 8? (CAT, 2012)
30
Find the remainder of 21040 divided by 131. (CAT, 2012)
31
A three-digit number which on being subtracted from another three-digit number consisting of the same digits in reverse order gives 594. The minimum possible sum of all the three digits of this number is (2013)
32
If a1=1 and a(n 1)-3an 2=4n for every positive integer n, then a100 equals ? (2014)
33
Suppose n is an integer, such that the sum of the digits of n is 2, and 10^10<N<10^11
34
A sequence of 4 digits, when considered as a number in base 10 is four times the number it represents in base 6. What is the sum of the digits of the sequence? (2016)
35
What is the greatest power of 5 that divides 80! exactly (2016)
36
If 92x-1- 81x-1= 1944, then x is (2017)
37
The right most non-zero digit of 30^2720 is? (2005)
38
Each family in a locality has at most two adults, and no family has fewer than three children. .. (2004)
39
Let x and y be positive integers such that x is prime and y is composite. (2003)
40
If sum of the first 11 terms of arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms? (2004)
41
The number of common terms in the two sequences 17, 21, 25, …, 417 and 16, 21, 26, …, 466 is (2008)
42
What is the remainder when 7 ^ 74 – 5 ^74 is divided by 4? (2008)
43
if a^2-ab b^2/a^2 ab b^2=1/3, then find (2009)
44
The largest number among the following that will perfectly… (2010)
45
How many integers, greater than 999 but not greater than 4000…(2010)
46
Find the smallest number which when increased by 5 is exactly divisible by 8, 11 and 24. (1994)
47
A student instead of finding the value of 7/8 of a number, found the value of 7/18 of the number (1997)
48
If n^3 is odd, which of the following statements are true? (1998)
49
What is the digit in unit’s place of 2^51? (1998)
50
If n=1+x, where x is the product of four consecutive positive integers, then which of the following is/are true? (1999)
51
The integers 34041 and 32506, when divided by a three digit integer n, leave the same remainder. What is the value of n? ( 2000)
52
Anita had to do a multiplication. Instead of taking 35 as one of the multipliers, she took 53. As a result, the product went up by 540. What is the new product? (2001)

Section 1

If sum of the first 11 terms of arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms? (2004)

 0 total views

 

If the sum of the first 11 terms of arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms?  [CAT, 2004]

a. 0     b. -1      c.1      d. Not Unique 

  DMCA.com Protection Status

Solution: 

 

Arithmetic Sequence or Arithmetic Progression (A.P)

 
Firstly understand that a sequence is a list of numbers that have a particular pattern. 

 

A sequence is also called arithmetic sequence or arithmetic progression.
 
For Example (1)  2, 4, 6, 8… is a sequence of even numbers.
                          (2)  4, 16, 36, 64… is a sequence of the squares of even numbers. 
In the above examples, (1) the next number 4 is found by adding a common difference (d), that is 2 in the example, to the previous number (called term) 2.
{ a }_{ n+1 }={ a }_{ n-1 }+d
where 
{ a }_{ n+1 }=4,   the next term
{ a }_{ n-1 }=2,   the previous term and
 d=2 which is the common difference.

 

Similarly, if we want to find the fourth term of this sequence we can use this formula.
{ a }_{ n }={ a }_{ 1 }+(n-1)d   
 where  { a }_{ 1 } is the first term of the sequence and n is the number of the term we want to find out.
{ a }_{ 4 }={ 2 }_{ 1 }+(4-1)2
{ a }_{ 4 }={ 2 }+(4-1)2
{ a }_{ 4 }={ 2 }+3\times 2
{ a }_{ 4 }={ 2 }+6
{ a }_{ 4 }=8.

 

For example, if we are considering only six terms in this sequence, 2, 4, 6, 8, 10, 12; and we want to find the sum of all the terms in the sequence. Then we can use the following formula:
{ S }_{ n }=\frac { n }{ 2 } [2a+(n-1)d]
where a is the first term and n is the number of terms in the sequence. Here n=6
{ S }_{ 6 }=\frac { 6 }{ 2 } [2\times 2+(6-1)2]
{ S }_{ 6 }=\frac { 6 }{ 2 } [4+(12-2)]
{ S }_{ 6 }=\frac { 6 }{ 2 } [4+10]
{ S }_{ 6 }=\frac { 6 }{ 2 } \times 14
{ S }_{ n }=42
 
If we add all the six terms of the given sequence, we find that their sum is 2+4+6+8+10+12=42.

 

Now let’s solve the given question in CAT:

 

 If sum of the first 11 terms of arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms?
 
Notice that the first term “a” is not given here nor have we been told about anything of the terms of the sequences. This is because the sum of both the sequences is told to be equal and because of this we have to make an equation and show that left-hand side (LHS) is equal to the right-hand side (RHS).

 

The formula for finding the sum of n terms of A.P: (as shown above)

 

{ S }_{ n }=\frac { n }{ 2 } [2a+(n-1)d]

 

Now put this sequence into the form of an equation and equate LHS and RHS.

 

\frac { 11 }{ 2 } [2a+(11-1)d]=\frac { 19 }{ 2 } [2a+(19-1)d]
 
Now solve the above equation:
 
\frac { 11}{ 2 } [2a+10d]=\frac { 19 }{ 2 } [2a+18d]

 

 \frac { 11}{ 2 } [2a+10d]=\frac { 19 }{ 2 } [2a+18d]

 

 \frac { 11 }{ 2 } \times 2a+10d=\frac { 19 }{ 2 } \times 2a+18d

 

 \frac { 22a+110d }{ 2 } =\frac { 38a+342d }{ 2 }

 

 22a+110d=38a+342d.

 

Now,  we will put zero on the left-hand side because we are saying that LHS=RHS.

 

We have to put a zero before transposing numbers from one hand side to other whenever we suppose that LHS=RHS  i.e.,   0=RHS-LHS.

 

     0=38a+342d-22a-110d
     0=38a-22a+342d-110d
     0=16a+232d
On solving the above equation, we find 0=16a+232d……(1)

 

Notice: We, after equating both sides, have found 16a+232d=0. Now, after putting the formula to find the sum of the first 30 terms ( which is the second part of the question), we have to make that equation to look like 16a+232d. Only then a relation between two parts of the question can be established.

 

Now if we put the second part of the question into the formula that says, then what is the sum of the first 30 terms.

 

{ S }_{ n }=\frac { n }{ 2 } [2a+(n-1)d]
{ S }_{ n }=\frac { 30 }{ 2 } [2a+(30-1)d]
{ S }_{ n }=\frac { n }{ 2 } [2a+(30d-d)]
{ S }_{ n }= 15(2a+29d)…….(2)

 

If we multiply (2a+29d) by 8, it becomes 16a+29d. This is what we found in the first part of the question and 16a+29d was equal to 0. This means when we find 16a+29d in the next part, we will make it equal to 0. 

 

In the given question a relation has been made that if sum of the first 11 terms of arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms.

 

This means after solving the first part of the question, 0=16a+232d, we have to use it in the second part of the question, Sn=15(2a+29d).

 

Since 16a+232d is not equal to (2a+29d), we have to make 16a+232d equal to 2a+29d, only then a relationship can be made between the two parts of the question.

 

We can find that either we divide 16a+232d by 8 it becomes 2a+29d or we can multiply 2a+29d by 8 and make it 16a+232d.

 

(Note: We are dividing by 8 because  16+232 are divisible by 8.)  Click here to know the divisibility rule of 8.
\frac { 16a+232d }{ 8 }  and it  is equal to zero     0=2a+29d as proved before in eq (1).

 

So, now putting the equation 2 :

 

{ S }_{ n }=15(2a+29d)…….(2)

 

since(2a+29d)=0 or (16a+232d)=0,  and    (2a+29d)= (16a+232d)=0
0= 15 (2a+29d),                       where (2a+29d) =0 

 

0= 15 x 0

 

Hence the option (a) is correct and the answer is 0.

 

 To know why a number multiplied by 0 is zero? click here.

error: Content is protected !!