Question

5th of an arithmetic sequence is 16 and its 8th term is 25. a)what is its common difference? b)what is it 6th term? c)what is the sum of its first 10 terms?

Answer 1

Answer:

(a) common difference is 3

(b) 6th term is 19

(c) sum of first 10 terms is 175

Step-by-step explanation:



Let, 1st term of an arithmetic sequence is a and the common difference is d

Then, we know that, n-th term of the sequence will be a+(n-1)d

5th term of the sequence is 16 hence, a+(5-1)d = 16 => a+4d = 16 --------(1)

8th term of the sequence is 25 hence, a+(8-1)d = 25 => a+7d = 25 --------(2)

From (2)-(1), we get, 3d = 9 => d = 3

Hence, (a) common difference is 3

Putting d =3 in equation (1) , we get, a + 12 = 16 => a = 4

(b) The 6th term will be 4+(6-1)3 = 4+15 = 19

(c) we know that , sum of first n terms of an arithmetic sequence is n/2 {2a +(n-1)d}

The sum of first 10 terms is = 10/2 {8+(10-1)3} = 5(8+27) = 5×35 = 175

Answer 2

Given:

5th of an arithmetic sequence is 16 and its 8th term is 25.

To Find:

a)what is its common difference? b)what is it 6th term? c)what is the sum of its first 10 terms?

Solution:

An arithmetic sequence is a sequence in which every term differs by some common difference which is denoted by 'd' and the first term of the sequence is denoted by 'a'.

The nth term of an AP is given by,

$T_n=a+(n-1)d$

And the sum of n terms of an AP is given by,

$S_n=\frac{n}{2}[2a+(n-1)d]$

Now,

(a) the common difference of an AP can be found by calculating the difference between any two terms but as here we are not given directly the sequence, we will first solve the given data that is,

$T_5=a+(5-1)d\\16=a+4d$

And,

$T_8=a+(8-1)d\\25=a+7d$

Now subtracting both the equations we have,

$25-16=a+7d-a-4d\\9=3d\\d=3$

So the common difference is 3.

Hence, the common difference is 3.

(b) The 6th term of the series will be,

As we have the 5th term as 16 and we know a common difference, so the 6th term will be,

$T_6=T_5+d\\=16+3\\=19$

Hence, the 6th term is 19.

(c) the sum of the first 10 terms will be,

As we know a common difference is 3 so the first term will be,

$16=a+4*3\\a=16-12\\a=4$

Now using the formula for the sum of AP and first term as 4, we have

$S_{10}=\frac{10}{2}[2*4+(10-1)*3]\\=5(8+27)\\=5*35\\=175$

Hence, the sum of the first 10 terms is 175.