Find the common difference of an AP in which a18 - a14 = 32

Arithmetic Progression(A.P) is a sequence of numbers with a constant difference between consecutive terms. This difference is called the Common Difference of the AP. It is denoted as 'd'. The first term of the AP is denoted as 'a'. 

For Example, 1, 5, 9, 13 is an AP having first term, a = 1 and common difference, d = (5-1) = 4.

Formula for the nth term of an AP

Suppose we have an AP, whose

• First term = a,
• Common difference = d.

Therefore, its

• Second term, a₂ = First Term + d = a + d

Similarly, 

• Third term, a₃ = Second term + d = (a + d) + d = a + 2d

Observing the pattern, 

The formula for nth term: 

a_n = a + (n - 1) × d

Formula for the Sum of an AP 

We have an AP having 'n' number of terms whose, 

First term = a, and 

Common difference = d.

For this AP, the formula for the Sum is, 

S_n = (n/2) × (a + l) 

where l is the last term and 

l = (a + (n - 1) × d)

Another formula for sum is, 

S_n = (n/2) × (2a + (n-1) × d) 

where we have substituted the value of l in terms of a and d.

Find the common difference of an AP in which a₁₈ - a₁₄ = 32

Answer:

Since, a_n = S_n - S_n-1 

i.e. if we subtract the Sum of n terms of an AP from the Sum of (n-1) terms of the same AP, we get the nth term of the AP.

So, a₁₈ = S₁₈ - S₁₇   .....(1)

Similarly, a₁₄ = S₁₄ - S₁₃   ....(2)

Let the first term of the AP be a and the common difference be d.

Putting (1) and (2) into the given equation: a₁₈ - a₁₄ = 32, we get

(S₁₈ - S₁₇) - (S₁₄ - S₁₃) = 32

(18/2)(2 × a + (18-1) × d) - (17/2)(2 × a + (17-1) × d) - (14/2)(2 × a - (14-1) × d) + (13/2)(2 × a + (13-1) × d) = 32

Solving this equation we get d = 8.

Hence, the Common Difference of the AP in which a₁₈ - a₁₄ = 32 is 8.

Another Approach:

As we know that, a_n = a + (n-1)Xd

a₁₈ - a₁₄ = 32

a + 17 x d - (a + 13 x d) = 32

a + 17xd - a - 13xd = 32

4xd = 32

therefore, common difference: d = 8

Similar Questions

Question 1: In an AP, a₂=4, and a₄=10. Find the common difference(d) of the AP.

Answer: 

Since,

a₂=a+d=4   ....(1)

a₄=a+3 × d=10   ....(2)

Subtracting (2) from (1) i.e. a₄ - a₂

a + 3 x d - (a + d) = 10 - 4

2 × d = 6

Hence, d = 3.

Question 2: In an AP, a₆=4, and a₈=1. Find the common difference(d) of the AP.

Answer: 

Since, 

a₆ = a+5 × d=4   ....(1)

a₈ = a+7 × d=1   ....(2)

Subtracting (2) from (1) i.e. a₈ - a₆

a + 7 x d - (a + 5 x d) = 4 -1

2 × d = -3

Hence, d = -1.5

Question 3: In an AP, a₃=14, and a₂=10. Find the first term(a) of the AP.

Answer: 

Since, 

a₃ = a+2 × d=14   ....(1)

a₂ = a+d=10   ....(2)

Subtracting (1) from (2) i.e.  a₃ - a₂

a + 2 x d - (a + d) = 14 - 10

d = 4

Putting d=4 in equation (2), we get

a + d = 10 

a = 10-4 

a= 6

Hence, the first term is 6.

Question 4: In an AP, a₁₄=4, and a₇=11. Find the first term(a) of the AP.

Answer: 

Since,

a₁₄=a+13 × d=4   ....(1)

a₇=a+6 × d=11   ....(2)

Subtracting (2) from (1) as,

7 × d = -7

d = -1

Putting d=-1 in equation (2), we get

a = 10 - d 

= 10 - (-1) 

= 10+1 

= 11

Question 5: Given the first and the last term of an AP with 10 terms are 10 and 40 respectively. Calculate the sum of the AP.

Answer: 

Since, 

a = 10, l = 40, n = 10

Sum = (n/2) × (a+l)

= (10/2) × (10+40) 

= 5 × 50 

= 250

Question 6: Given the first term of an AP with 10 terms and the common difference of 10, is 10. Calculate the sum of the AP.

Answer: 

Since, 

a = 10, d = 10, n = 10

Sum = (n/2) × (2 × a+(n-1) × d)

= (10/2) × (20+9 × 10)

= 5 × (20+90) 

= 5 × 110 

= 550.

Practice Problems

1. Find the first term if the common difference of an AP is 5 and a₁₀=45.
2. Determine the number of terms in an AP where the first term is 3 the common difference is 7 and last term is 66.
3. If the sum of the first 15 terms of an AP is 300 and common difference is 4 find the first term.
4. Find the 12th term of an AP if the first term is 2 and common difference is 3.
5. The 5th term of an AP is 15 and 10th term is 35. Find the common difference.
6. In an AP, the 7th term is twice the 4th term. Find the common difference.
7. If the sum of the first 20 terms of an AP is 210 find the common difference when the first term is 2.
8. The 3rd term of an AP is 7 and 7th term is 15. Find the 10th term.
9. If the common difference of an AP is 6 and the first term is 1 find the sum of the first 12 terms.
10. Find the common difference if the sum of the first 8 terms of an AP is 72 and first term is 2.

• Arithmetic Progression
• Arithmetic Progression and Geometric Progression (AP And GP)
• Arithmetic Progressions Class 10: NCERT Notes