Question

# The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, the number of terms is

- 10
- 11
- 12
- None of these

Hint:

### The difference between any two consecutive integers in an arithmetic progression (AP) sequence of numbers is always the same amount. It also goes by the name Arithmetic Sequence. Here we have given that the sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, we have to find the number of terms.

## The correct answer is: 11

### A progression of numbers known as an arithmetic sequence is one in which, for every pair of consecutive terms, the second number is derived by adding a predetermined number to the first one.

There are three types of progressions in mathematics. As follows:

- Arithmetic Progression (AP)
- Geometric Progression (GP)
- Harmonic Progression (HP)

In AP, we will come across some main terms, which are denoted as:

- First term (a)
- Common difference (d)
- nth Term (a
_{n})
- Sum of the first n terms (S
_{n})

Here we have given sum of the first four terms of an A.P. is 56 and sum of the last four terms is 112. Then:

Let consider the A.P. as:

*a*, *a* + *d*, *a* + 2*d*, *a* + 3*d*, ... *a* + (*n* – 2) *d*, *a* + (*n* – 1)*d*.

Then the sum of first four terms will be:

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

4*a* + 6*d*

The sum of last four terms will be:

[*a* + (*n* – 4) *d*] + [*a* + (*n* – 3) *d*] + [*a* + (*n* – 2) *d*] + [*a* + *n* – 1) *d*]

4*a* + (4*n* – 10) *d*

*Lets take the first condition as sum is 56, we get:*

4*a* + 6*d* = 56

We have given a=11, so we get:

4(11) + 6*d* = 56

6*d* = 12

*d* = 2

So therefore, we have:

4*a* + (4*n* –10) *d* = 112

4(11) + (4*n* – 10)x2 = 112

(4*n* – 10)x2 = 68

4*n* – 10 = 34

4*n* = 44

*n* = 11

So therefore the number of terms in this AP is 11.

_{n})_{n})Here we used the concept of Arithmetic progression to find the number of terms in the given series. When we study Arithmetic Progression, which is associated with: There are two key formulas we encounter, those were nth term of AP and sum of the first n terms. So therefore the number of terms in this AP is 11.

### Related Questions to study

### The sum of all two digit numbers which when divided by 4 leaves 1 as remainder is

Here we used the concept of Arithmetic progression to find the number of terms in the given series. When we study Arithmetic Progression, which is associated with: There are two key formulas we encounter, those were nth term of AP and sum of the first n terms. So therefore the sum is 1210.

### The sum of all two digit numbers which when divided by 4 leaves 1 as remainder is

Here we used the concept of Arithmetic progression to find the number of terms in the given series. When we study Arithmetic Progression, which is associated with: There are two key formulas we encounter, those were nth term of AP and sum of the first n terms. So therefore the sum is 1210.