The 5th term of an arithmetic sequence is 22 and the 15th term is 62. Find the 100th term and the sum of the first 60 terms.

see related questions below

To find the 100th term of an arithmetic sequence, we need to know the common difference.

First, we need to find the common difference (d) of the arithmetic sequence. We can do this by subtracting the 5th term from the 15th term.

15th term - 5th term = 62 - 22 = 40

Now that we know the common difference, we can find the 100th term using the formula for the nth term of an arithmetic sequence:

nth term = first term + (n - 1) * common difference

100th term = 5th term + (100 - 5) * common difference
= 22 + 95 * 40
= 22 + 3800
= 3822

Therefore, the 100th term of the arithmetic sequence is 3822.

To find the sum of the first 60 terms of an arithmetic sequence, we can use the formula for the sum of an arithmetic sequence:

sum of n terms = (n/2) * (2 * first term + (n - 1) * common difference)

We can plug in the values into the formula:

sum of 60 terms = (60/2) * [2 * first term + (60 - 1) * common difference]

But we only have the information about the 5th term, so we need to find the first term (a).

To find the first term, we can subtract (n - 1) * common difference from the nth term.

First term = nth term - (n - 1) * common difference

First term = 22 - (5 - 1) * 40
= 22 - 4 * 40
= 22 - 160
= -138

Now we can find the sum of the first 60 terms:

sum of 60 terms = (60/2) * [2 * (-138) + (60 - 1) * 40]
= 30 * (-2760 + 59 * 40)
= 30 * (-2760 + 2360)
= 30 * (-400)
= -12000

Therefore, the sum of the first 60 terms of the arithmetic sequence is -12000.

To find the 100th term of an arithmetic sequence, we need to know the common difference. Since the arithmetic sequence is not given explicitly, we can use the given information to determine the common difference.

We know that the 5th term of the sequence is 22, which means that the equation for the 5th term in terms of the common difference is:

a + 4d = 22

Similarly, the equation for the 15th term in terms of the common difference is:

a + 14d = 62

where 'a' is the first term of the sequence and 'd' is the common difference.

Now, we have a system of two equations with two variables.
Using any method to solve the system, such as substitution or elimination, we can find the values of 'a' and 'd'.

Let's use elimination method:

Multiply the first equation by 14 and the second equation by 4 to get rid of the variable 'a'.

14(a + 4d) = 14(22) => 14a + 56d = 308

4(a + 14d) = 4(62) => 4a + 56d = 248

Next, subtract the second equation from the first equation:

(14a + 56d) - (4a + 56d) = 308 - 248

10a = 60

Divide both sides by 10:

a = 6

Substitute the value of 'a' back into either of the original equations to find 'd':

6 + 4d = 22

4d = 22 - 6

4d = 16

d = 4

So, the first term 'a' is 6 and the common difference 'd' is 4.

Now, to find the 100th term of the sequence:

The formula to find the nth term of an arithmetic sequence is:

nth_term = a + (n - 1) * d

Substituting the given values into the formula:

100th term = 6 + (100 - 1) * 4

100th term = 6 + 99 * 4

100th term = 6 + 396

100th term = 402

Therefore, the 100th term of the arithmetic sequence is 402.

To find the sum of the first 60 terms, we can use the formula for the sum of an arithmetic sequence:

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

Substituting the given values into the formula:

sum = (60/2) * (2 * 6 + (60 - 1) * 4)

sum = 30 * (12 + 59 * 4)

sum = 30 * (12 + 236)

sum = 30 * 248

sum = 7440

Therefore, the sum of the first 60 terms of the arithmetic sequence is 7440.

d=term difference÷position difference

62-22÷15-5=40÷10=4
d=4
tn=t1+(n-1)d
t100=6+100-1×4
=6+99×4=402 t1=t5-4d
=22-4×4
=22-16
=6