The 7 term and 20 term of an ap ate 34 and 112 respectively find the ap

Question

The 7 term and 20 term of an ap ate 34 and 112 respectively find the ap

Answer 1

Let's assume that the first term of the arithmetic progression is 'a', and the common difference is 'd'.

We know that the seventh term is 34, so we can write it as:
a + 6d = 34 ...(1)

We also know that the twentieth term is 112, so we can write it as:
a + 19d = 112 ...(2)

To find the arithmetic progression, we need to solve these equations simultaneously.

From equation (1), we can rewrite it as:
a = 34 - 6d

Substituting this value of 'a' into equation (2):
34 - 6d + 19d = 112
13d = 78
d = 6

Substituting the value of 'd' into equation (1):
a + 6(6) = 34
a + 36 = 34
a = -2

Therefore, the first term of the arithmetic progression is -2 and the common difference is 6. So, the arithmetic progression is: -2, 4, 10, 16, 22, 28, 34, ...

Answer 2

To find the arithmetic progression (AP), we can use the formula for the nth term of an AP:

an = a + (n-1)d

Where:
an = nth term of the AP
a = first term of the AP
n = position of the term in the AP
d = common difference between consecutive terms

Given that the 7th term (a7) is 34 and the 20th term (a20) is 112, we can set up the following equations:

a7 = a + 6d ......... (1)
a20 = a + 19d ........ (2)

To solve for a and d, we can use these equations.

Step 1: Solve Equation (1) for a in terms of d
a = a7 - 6d

Step 2: Substitute the value of a from Step 1 into Equation (2)
a20 = a7 - 6d + 19d
112 = a7 + 13d

Step 3: Substitute the value of a7 from the given information
112 = 34 + 13d

Step 4: Solve for d
112 - 34 = 13d
78 = 13d
d = 78/13
d = 6

Step 5: Substitute the value of d back into Equation (1) to solve for a
a7 = a + 6d
34 = a + 6*6
34 = a + 36
a = 34 - 36
a = -2

Step 6: Verify the values of a and d using Equation (2)
a20 = a + 19d
a20 = -2 + 19*6
a20 = -2 + 114
a20 = 112

Therefore, the AP is -2, 4, 10, 16, 22, 28, 34, 40, 46, 52, ....

Answer 3

To find the arithmetic progression (AP), we can use the following formula:

𝑡ₙ = 𝑎 + (𝑛 − 1)𝑑

Where:
𝑡ₙ = the nth term of the AP
𝑎 = the first term of the AP
𝑑 = the common difference between terms
𝑛 = the number of terms

Given that the 7th term is 34 and the 20th term is 112, we can use these values to solve for 𝑎 and 𝑑.

Step 1: Finding the common difference (𝑑)
Using the formula for the 7th term, we have:

34 = 𝑎 + (7 − 1)𝑑
34 = 𝑎 + 6𝑑

Step 2: Finding the first term (𝑎)
Using the formula for the 20th term, we have:

112 = 𝑎 + (20 − 1)𝑑
112 = 𝑎 + 19𝑑

Now, we have a system of two equations with two unknowns:

34 = 𝑎 + 6𝑑 --(1)
112 = 𝑎 + 19𝑑 --(2)

To solve this system of equations, we can use different methods such as substitution or elimination. Let's use the elimination method.

Step 3: Solving the system of equations
Multiply equation (1) by 19 and equation (2) by 6 to make the coefficients of 𝑑 in both equations equal:

(19) * (34) = (19)(𝑎 + 6𝑑)
646 = 19𝑎 + 114𝑑 --(3)

(6) * (112) = (6)(𝑎 + 19𝑑)
672 = 6𝑎 + 114𝑑 --(4)

Now subtract equation (3) from equation (4) to eliminate 𝑎:

672 - 646 = 6𝑎 + 114𝑑 - 19𝑎 - 114𝑑
26 = -13𝑎

Divide both sides by -13:

𝑎 = -2

Step 4: Substituting 𝑎 into either equation (1) or (2) to solve for 𝑑
Let's substitute 𝑎 = -2 into equation (1):

34 = -2 + 6𝑑
36 = 6𝑑

Divide both sides by 6:

𝑑 = 6

Step 5: Writing the arithmetic progression (AP)
Now that we have found the values of 𝑎 and 𝑑, we can write the AP:

𝑡ₙ = 𝑎 + (𝑛 − 1)𝑑

𝑡ₙ = -2 + (𝑛 − 1)(6)
𝑡ₙ = -2 + 6𝑛 - 6
𝑡ₙ = 4 + 6𝑛

The arithmetic progression is given by 𝑡ₙ = 4 + 6𝑛.