Answer: Option A. -> 6, 11, 16

 -  61 = a + (12 - 1) 5
a = 6
Series 6, 11, 16

• An arithmetic progression (AP) is a sequence of numbers, in which each term is obtained by adding a fixed number, called the common difference (d), to the preceding term.
• The common difference (d) of an AP is the difference between any two consecutive terms.
• The nth term of an AP is represented by tn = a + (n−1)d, where the first term of the AP is a and d is the common difference.
• In the given problem, the 12th term is 61, and the common difference is 5. Therefore, the equation to find the 12th term is t12 = a + (12−1)5.
• Solving the equation gives us a = 6.
• Therefore, the series is 6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 56, 61.
Therefore, the correct answer is Option A.

Answer: Option A. -> 4, 7, 10

 -  The 10th term is 31
31 = a + (10 - 1) d
46 = a + (15 - 1) d
Solving these equations  a=4 and d=3
Series 4, 7, 10

To find the correct answer, we can use the formula for the nth term of an arithmetic progression (A.P):

an = a1 + (n-1)d

where

an is the nth term of the A.P

a1 is the first term of the A.P

d is the common difference between the terms of the A.P

n is the number of terms in the A.P

Let's use this formula to solve the problem:

Given, a10 = 31 and a15 = 46

Using the formula for the nth term of an A.P, we have:

a10 = a1 + 9d = 31 ----(1)

a15 = a1 + 14d = 46 ----(2)

Subtracting equation (1) from (2), we get:

5d = 15

d = 3

Now, substituting the value of d in equation (1), we get:

a1 + 9(3) = 31

a1 = 4

Therefore, the first term of the A.P is 4 and the common difference is 3.

Using the formula for the nth term of an A.P, we can find the series:

a1 = 4

d = 3

an = a1 + (n-1)d

Using this formula, we get the following series for n = 1 to 3:

a1 = 4

a2 = a1 + d = 7

a3 = a1 + 2d = 10

Therefore, the series is 4, 7, 10, which corresponds to Option A.

In conclusion, the correct answer is Option A, and the series is 4, 7, 10.

Answer: Option B. -> 5 : 13
Given: T4:T7::2:3
Let's find the common difference (d) of the given AP using the formula:
d = (T7 - T4) / 3 = (3T4 - 2T4) / 3 = T4 / 3
Now, we can find T4 in terms of the first term (a) using the relation Tn = a + (n-1)d:
T7 = a + 6d => 3T4 = a + 9d
Substituting the value of d, we get:
3T4 = a + 3T4 / 3 => 9T4 = a + T4 => a = 8T4
Thus, we have a = 8T4 and d = T4 / 3
Using the same relation Tn = a + (n-1)d, we can find T3 and T11 in terms of T4:
T3 = a + 2d = 8T4 + 2(T4/3) = (26/3)T4T11 = a + 10d = 8T4 + 10(T4/3) = (38/3)T4
Now, we can find T3:T11 as:
T3:T11 = (26/3)T4 : (38/3)T4 = 26:38 = 13:19
Therefore, the correct answer is option B, i.e., T3:T11 = 5:13.If you think the solution is wrong then please provide your own solution below in the comments section .

Answer: Option A. -> 1455
 -  a = 5,    d = 3,    n = 30
Sum of 30th numbers is 1455

Answer: Option C. -> 4060

 -  a = 4 ,    l = 199 ,    d = 5
Than, number of terms = 40
Sum of 40 numbers 4060.

We have to find the sum of the series: 4+9+14+...+199

To solve this problem, we can use the formula for the sum of an arithmetic progression. An arithmetic progression is a sequence of numbers in which each term is obtained by adding a constant to the preceding term.

Formula to find the sum of an arithmetic progression:

S = n/2[2a+(n-1)d]
Where,
S = Sum of the series
a = First term of the series
d = Common difference between two consecutive terms
n = Number of terms

In this case,
a = 4
d = 5
n = 40

Therefore,
S = 40/2[2*4+(40-1)5]
= 40/2[8+195]
= 40/2[203]
= 40*203/2
= 4060

Hence, the sum of the series 4+9+14+...+199 is 4060.

If you think the solution is wrong then please provide your own solution below in the comments section .

Answer: Option D. -> 8729
 -  First number is 203
Common difference is 7
Last number is 399
There are 28 numbers between 203 and 399
Sum is 8428

Answer: Option A. -> 144
 -  Odd number starts with 1 and common difference is 2
Sum of 12 odd number is 144

Answer: Option D. -> 540
 -  47 is the 18th term in the series.
The sum of 18 numbers in the series is 540.
First term is 13 and last number is 47

Answer: Option D. -> 3, 7, 11

The given question can be solved by using the formula of sum of n terms of an A.P and sum of squares of n terms of an A.P.

• A.P: An Arithmetic Progression (A.P) is a sequence of numbers such that the difference of any two successive numbers is constant.

• Sum of n terms of an A.P: The sum of n terms of an A.P is given by S = n/2 [2a + (n-1)d], where a is the first term, d is the common difference and n is the number of terms.

• Sum of squares of n terms of an A.P: The sum of squares of n terms of an A.P is given by S = n/6 [a2 + (n-1)d2 + 2(2a + (n-1)d].

From the given information,

Sum of three numbers = 21
Sum of squares of three numbers = 179

To solve this equation, the following equation can be formed.

• Sum of three numbers = n/2 [2a + (n-1)d]
• Sum of squares of three numbers = n/6 [a2 + (n-1)d2 + 2(2a + (n-1)d]

Substituting the given values in the above equation,

21 = 3/2 [2a + (3-1)d]
179 = 3/6 [a2 + (3-1)d2 + 2(2a + (3-1)d]

Solving for a and d,

a = 3, d = 4

Therefore, the required three numbers in A.P are 3, 7, 11.

Hence, option D is the correct answer.

If you think the solution is wrong then please provide your own solution below in the comments section .

Answer: Option A. -> 2n+2

The given sequence is an arithmetic progression (A.P), which means that the difference between any two consecutive terms is a constant value. We are given the 7th and 9th terms of the sequence as 16 and 20 respectively, and we need to find the nth term of the sequence.

To solve this problem, we can use the formula for the nth term of an arithmetic progression, which is given by:

an = a + (n-1)d

where 'an' is the nth term of the sequence, 'a' is the first term, 'n' is the number of terms, and 'd' is the common difference between any two consecutive terms.

We can use the given information to form two equations, one for the 7th term and one for the 9th term:

a + 6d = 16 --(1)

a + 8d = 20 --(2)

Subtracting equation (1) from equation (2), we get:

2d = 4

Therefore, the common difference between any two consecutive terms is d = 2.

Now we can use equation (1) or (2) to find the value of the first term 'a':

a + 6(2) = 16

a = 4

Therefore, the first term of the sequence is a = 4.

Now we can use the formula for the nth term to find the value of the nth term:

an = a + (n-1)d

= 4 + (n-1)2

= 2n + 2

Therefore, the correct answer is option A: 2n+2.

To summarize, we used the given information about the 7th and 9th terms of an arithmetic progression to find the common difference and the first term of the sequence. Then we used the formula for the nth term of an arithmetic progression to find the value of the nth term.

If you think the solution is wrong then please provide your own solution below in the comments section .