Question

The sum
first 7 terms of an arithmetic sequences and write and 19 and the sum of first 20 terms is 860 what is its fourth term

Answer 1

The 4th term of the A.P. is 17.

The 17th term of the A.P. is 69.

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Let's understand a few concepts:

To solve the given problem we will use the following formulas:

• The sum of the first n terms of an A.P. can be calculated as:

               

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

• The nth term of an A.P. can be calculated as:

             $\boxed{\bold{a_n = a + (n-1)d}}$

where a = first term of the A.P., d = common difference and n = no. of terms

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Let's solve the given problem:

(a) Finding the 4th term of the A.P.:

The sum of the first 7 terms of an A.P. = 119

∴ $S_7 = \frac{7}{2} [2a + (7-1)d] = 119$

$\implies \frac{7}{2} [2a + 6d] = 119$

$\implies 2a + 6d = 119 \times \frac{2}{7}$

$\implies 2a + 6d = 34$

$\implies a + 3d = 17 \:.\:.\:.\:(1)$

We know,

$a_4 = a + (4 - 1)d$

$\implies a_4 = a + 3d$

from (1), we get

$\implies a_4 = a + 3d = 17$

$\implies \bold{a_4 = 17}$

Thus, the 4th term of the A.P. is 17.

(b) Finding the 17th term of the A.P.:

The sum of the first 20 terms of the A.P. = 860

∴ $S_2_0 = \frac{20}{2} [2a + (20-1)d] = 860$

$\implies \frac{20}{2} [2a + 19d] = 860$

$\implies 10[2a + 19d] = 860$

$\implies 2a + 19d = 86\:.\:.\:.\:(2)$

On multiplying equation (1) by 2, we get

$a + 3d = 17 ]\times 2\\\implies2a + 6d = 34\:.\:.\:.\:(3)$

On subtracting equation (3) from (2), we get

$2a + 19d = 86\\\\2a + 6d = 34\\\\-\:\:\:-\:\:\:\:\:\:-\\-------\\\\13d = 52\\-------$

∴ $d = \frac{52}{13} = 4$

On substituting d = 4 in equation (1), we get

$a + (3\times 4) = 17$

$\implies a + 12 = 17$

$\implies a = 5$

Now,

The 17th term of the A.P. is,

= $\bold{a_1_7}$

= $a + (17 - 1)d$

= $a + 16d$

= $5 + (16\times 4)$

= $5 + 64$

= $\bold{69}$

Thus, the 17th term of the A.P. is 69.

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Complete Question:

The sum of the first 7 terms of an arithmetic sequence is 119 and the sum of the first 20 terms is 860.

a)What is the 4th term?

b)What is its 17th term?

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