Question

1. The sum of 4th and 6th progression is 42. The sum of 3rd and 9th term of the progression is 52. Find a). First term b.) the common difference. c). the sum of the first 10 terms of the progression.​

Answer 1

Hence,

Common difference is

First term is

Sum of first 10 terms is 47

Explanation:

a4 + a6 =42 eq1

a3 + a9 =52 eq

a1=?

d=?

S10=?

From eq 1,

a4 + a6=42

a1+3d + a1+5d =42

2a1 + 8d= 42 eq 3

From eq 2,

a3 + a9 =52

a1+2d + a1+8d = 52

2a1 + 10d = 52 eq 4

Subtract eq 3 from eq

2a1 + 10d =52

-2a1 -8d = -42

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2d = 10

d=5....

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Putting value of d in eq

2a1 +10 d=52

2a1+ 50 =52

a1 =1

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Now we find sum of first 10 terms,

S10 = 2a1 + 9d

S10 = 2+45

S10 = 47

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Answer 2

a) The first term of the progression is 4.5.

b) The common difference of the progression is 4.

c) The sum of the first 10 terms of the progression is 225.

Explanation:

To solve this problem, you can use the formula for the sum of an arithmetic series. An arithmetic series is a sequence of numbers in which each term is obtained by adding a fixed number, called the common difference, to the preceding term.

The sum of an arithmetic series with n terms and a common difference d is given by:

Sum = n/2 * (2a + (n-1)d)

Where a is the first term of the series and d is the common difference.

In this problem, you are given that the sum of the 4th and 6th terms is 42 and the sum of the 3rd and 9th terms is 52. You can use these two equations to solve for a and d.

First, let's find the sum of the 4th and 6th terms:

Sum = 4/2 * (2a + 3d) = 42

This simplifies to:

2a + 3d = 21

Next, let's find the sum of the 3rd and 9th terms:

Sum = 6/2 * (2a + 5d) = 52

This simplifies to:

2a + 5d = 26

Now that we have two equations, we can solve for a and d by using substitution or elimination.

To use substitution, we can solve the second equation for d:

d = (26 - 2a)/5

Then, we can substitute this expression for d into the first equation:

2a + 3((26 - 2a)/5) = 21

This simplifies to:

2a + 3(26 - 2a)/5 = 21

Which simplifies to:

2a + 3(26)/5 - 3(2a)/5 = 21

This simplifies to:

2a + 3(26)/5 - 6a/5 = 21

This simplifies to:

-4a + 3(26)/5 = 21

This simplifies to:

-4a + 39 = 21

This simplifies to:

-4a = -18

This simplifies to:

a = 4.5

Now that we know the value of a, we can substitute it back into one of the original equations to find the value of d:

2(4.5) + 3d = 21

This simplifies to:

9 + 3d = 21

This simplifies to:

3d = 12

This simplifies to:

d = 4

Now that we have found the values of a and d, we can use the formula for the sum of an arithmetic series to find the sum of the first 10 terms of the progression:

Sum = 10/2 * (2(4.5) + (10-1)(4))

= 10/2 * (9 + 36)

= 10/2 * 45

= 225

Therefore, the sum of the first 10 terms of the progression is 225.