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The 11th term of an progression is 25 and the sum of the first 4 terms is 49. The nth term of the progression is 491. Find the first term of the progression and the common difference
2. Find the value of n

Answers

Answer 1

Answer:

Answer:

For 1: The first term is 10 and the common difference is $(3)/(2)$

For 2: The value of n is 27

Step-by-step explanation:

The n-th term of the progression is given as:

$a_n=a_1+(n-1)d$

where,

$a_1$ is the first term, n is the number of terms and d is the common difference

The sum of n-th terms of the progression is given as:

$S_n=(n)/(2)[2a_1+(n-1)d]$

where,

$S_n$ is the sum of nth terms

For (1):

The 11th term of the progression:

$25=a_1+10d$ .......(1)

Sum of first 4 numbers:

$49=(4)/(2)[2a_1+3d$ ......(2)

Forming equations:

$98=8a_1+12d$

$25=a_1+10d$ ( × 8)

The equations become:

$98=8a_1+12d$

$200=8a_1+80d$

Solving above equations, we get:

$102=68d\n\nd=(102)/(68)=(3)/(2)$

Putting value in equation (1):

$25=a_1+10(3)/(2)\n\na_1=[25-15]=10$

Hence, the first term is 10 and the common difference is $(3)/(2)$

For 2:

The nth term is given as:

$49=10+(n-1)(3)/(2)$

Solving the above equation:

$39=(n-1)(3)/(2)\n\nn-1=26\n\nn=27$

Hence, the value of n is 27

Answer 2

Answer:

Final answer:

The value of n when the nth term of the progression is 49 is 22.

Explanation:

The 11th term of the progression (a11) is 25.

The sum of the first 4 terms (S4) is 49.

The nth term (an) is 49.

Let's find the answers to your questions:

Find the first term of the progression (a1) and the common difference (d):

We know that the nth term of an AP can be expressed as:

an = a1 + (n - 1)d

Substituting the values:

a11 = a1 + (11 - 1)d

25 = a1 + 10d

Now, we need to find a1 and d. We'll also use the information that the sum of the first 4 terms (S4) is 49. In an AP, the sum of the first n terms (Sn) can be expressed as:

Sn = (n/2)[2a1 + (n - 1)d]

For S4:

49 = (4/2)[2a1 + (4 - 1)d]

49 = 2[2a1 + 3d]

Now, we have two equations:

25 = a1 + 10d

49 = 2[2a1 + 3d]

Let's solve this system of equations to find a1 and d.

1. First, rearrange the first equation to isolate a1:

a1 = 25 - 10d

Now, substitute this expression for a1 into the second equation:

49 = 2[2(25 - 10d) + 3d]

Simplify and solve for d:

49 = 2[50 - 20d + 3d]

49 = 2[50 - 17d]

49 = 100 - 34d

34d = 100 - 49

34d = 51

d = 51/34

d = 3/2

2. Now that we have the common difference (d), we can find a1 using the first equation:

a1 = 25 - 10d

a1 = 25 - 10(3/2)

a1 = 25 - 15/2

a1 = (50 - 15)/2

a1 = 35/2

a1 = 17.5

So, the first term of the progression (a1) is 17.5, and the common difference (d) is 3/2.

Find the value of n when the nth term of the progression is 49:

We know that an = 49, and we can use the formula for an in an AP:

an = a1 + (n - 1)d

Substitute the values:

49 = 17.5 + (n - 1)(3/2)

49 - 17.5 = (n - 1)(3/2)

31.5 = (n - 1)(3/2)

To isolate n, multiply both sides by (2/3):

(n - 1)(3/2) = 31.5 * (2/3)

(n - 1) = 21

Now, add 1 to both sides to find n:

n = 21 + 1

n = 22

So, the value of n when the nth term of the progression is 49 is 22.

Learn more about Arithmetic Progression here:

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Answers

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Answers

Answer:

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Answers

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Answers

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Answers

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