Basic Adding Patterns in an AP

Q. If t(n) represents the $n^{th}$ term of an AP, then which of the following is always true?

1. t(1) + t(20) = t(19) + t(2)
2. t(2) + t(18) = t(4) + t(16)
3. t(1) + t(20) = t(21)
4. t(9) + t(10) = t(19)

Q.
In an AP, the sum of the first term and the last term is equal to the sum of the second term and the second last term and so on.

1. True
2. False

Q. Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be arithmetic progressions such that $a_1=25,b_1=75$ and $a_{100}+b_{100}=100$. Find the sum of the first one hundred terms of the progression $a_1+b_1,a_2+b_2,\dots$

1. 100
   
2. 10, 000
   
3. 505, 000
   
4. 0
   

Q.

If $a_1,a_2,a_3,a_4$ are in AP, then

1. $a_1+a_4=a_2+a_3$

2. $a_1+a_2=a_2+a_3$

3. $a_1+a_2=a_3+a_4$

4. $a_1+a_3=a_2+a_4$

Q.

Which of the following series forms an AP?

(i) a – 2d, a – d, a, a + d, a + 2d

(ii)a, a + d, a + 2d, a + 3d, a + 4d

(iii) a – 3d, a – d, , a + d, a + 3d

1. Only (ii) and (iii)

2. All (i), (ii) and (iii)

3. Only (i) and (iii)

4. Only (i) and (ii)

Q.

Find the sum of the first 10 natural numbers.
(Use $S_{10}=1+2+3+\dots +10$ and proceed)

1. $10\times 11$

2. $\frac{(10\times 11)}{2}$

3. $\frac{(10\times 2)}{11}$

4. None of these

Q. <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> The sum of the third and the seventh terms of an $AP$ is $6$ and their
product is $8$. Find the sum of first sixteen terms of the $AP$ .

Q. For an AP of 14 terms, find the value of $a_9$, if $a_4+a_{11}=118$ and $a_6=56$.

1. 62
2. 58
3. 70
4. 64

Q. For an AP of 14 terms, find the value of $a_9$, if $a_4+a_{11}=118$ and $a_6=56$.

1. 62
2. 58
3. 64
4. 70

Q. For an AP of 14 terms, find the option which gives the value equal to $a_4+a_{11}$.

1. $a_3+a_8$
2. $a_6+a_9$
3. $a_9+a_7$
4. $a_5+a_7$

Q. Question 12
Two APs have the same common difference. The difference between their ${100}^{th}$ terms is 100, what is the difference between their ${1000}^{th}$ terms?

Q. Sum of the first 10 natural numbers =

1. 110
2. 55
3. 45

Q. If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289 then the common difference of the AP is

1. 1
2. 2
3. 10
4. 3

Q.
In an AP, the sum of the first term and the last term is equal to the sum of the second term and the second last term and so on.

1. True
2. False

Q. The sum of the first $9$ terms of an $A.P$ is $81$ and the sum of it's first $20$ terms is $400.$ Find the first term, the common difference and the sum upto $15th$ term.

1. $a=1,d=2,$ $S_{15}=235$
2. $a=5,d=3,S_{15}=205$
3. $a=3,d=4,S_{15}=215$
4. None of these
   

Q. If $7^{th}$ and ${13}^{th}$ terms of an A.P. are 34 and 64, respectively, then its ${18}^{th}$ term is

1. $87$
2. $88$
3. $90$
4. $89$

Q. A number consist of two digits. When it is divided by the sum of its digit, the quotient is $6$ with no reminder. When the number is diminished by $9$, the digits are reversed. Find the number.

Q. Find the respective terms for the following $APs$
$a_1=5,a_4=9\frac{1}{2}$ find $a_2,a_3$

Q. There are $60$ terms in an A.P. of which the first term is $8$ and the last term is $185.$ The ${31}^{st}$ term is

1. $56$
2. $94$
3. $85$
4. $98$

Q. Find the common difference the following A.P
$1,4,7,10,13,16,..$

Q. If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289 then find the common difference of the AP.

1. 2
2. 1
3. 3
4. 10

Q. If the $5$th term of an A.P is $31$ and $25$th term is $140$ more than the $5^{th} term, find the first term of the A.P.

Q. If t(n) represents the $n^{th}$ term of an AP, then which of the following is always true?

1. t(1) + t(20) = t(19) + t(2)
2. t(2) + t(18) = t(4) + t(16)
3. t(1) + t(20) = t(21)
4. t(9) + t(10) = t(19)

Q. For a sequence of natural numbers find ∑n=201∑1n=20

Q.

If 9th term of an AP is 15 and the common difference is 2, the 1st term of the AP is

1. 1

2. 0

3. -1

4. 2

Q. If $a_1,a_2,a_3,a_4$ are terms of an AP, then the relation between them is

1. $a_1+a_2=a_2+a_3$
2. $a_1+a_4=a_2+a_3$
3. $a_1+a_3=a_2+a_4$

Q. In an AP where the number of terms is odd, the sum of the first and the last term will be twice the middle term.

1. False
2. True

Q. If the sum of $n$ terms of an AP is $\frac{n}{2}(7+5n)$, then its $n^{th}$ term is

1. $5n+1$
2. $7n+1$
3. $2n+1$
4. $5n+2$

Q. Find the sum of the two middle most terms of the AP : $-\frac{4}{3},-1,\frac{-2}{3},-\frac{1}{3},....,4\frac{1}{3}$.

Q.
In an AP, the sum of the first term and the last term is equal to the sum of the second term and the second last term and so on.

1. True
2. False