The first three terms of an arithmetic sequence are −5, k, 1. (i) Find k and hence show that the common difference is 3. (ii) Find T_{10}, the 10th term in the sequence. (iii) Find which term in the sequence has a value of 247. (b) The first three terms of a different arithmetic sequence are 4, 9, 14. Find S_{50}, the sum of the first 50 terms of the sequence.

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The first three terms of an arithmetic sequence are −5, k, 1 - Leaving Cert Mathematics - Question 6 - 2021

Question 6

The first three terms of an arithmetic sequence are −5, k, 1. (i) Find k and hence show that the common difference is 3. (ii) Find T_{10}, the 10th term in the se... show full transcript

Worked Solution & Example Answer:The first three terms of an arithmetic sequence are −5, k, 1 - Leaving Cert Mathematics - Question 6 - 2021

Step 1

Find k and hence show that the common difference is 3.

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Answer

In an arithmetic sequence, the difference between consecutive terms is constant. We can express this relationship as:

$T_2 - T_1 = T_3 - T_2$ Substituting the given terms:

$k - (-5) = 1 - k$ This simplifies to: $k + 5 = 1 - k$ Adding k to both sides gives: $2k + 5 = 1$ Next, subtract 5 from both sides: $2k = 1 - 5$ Which simplifies further to: $2k = -4$ Hence, we find: $k = -2$

Now, to find the common difference: The first term is -5 and the second term is -2.

Common difference: $d = -2 - (-5) = -2 + 5 = 3$ Thus, the common difference is confirmed as 3.

Step 2

Find T_{10}, the 10th term in the sequence.

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Answer

Using the formula for the nth term of an arithmetic sequence:

$T_n = T_1 + (n - 1)d$

For this sequence, the first term, $T_1 = -5$ , and the common difference $d = 3$ .

To find $T_{10}$ :

$T_{10} = -5 + (10 - 1)3$ $T_{10} = -5 + 9 = 4$ Therefore, $T_{10} = 22$ .

Step 3

Find which term in the sequence has a value of 247.

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Answer

We start again with our nth term formula:

$T_n = T_1 + (n - 1)d$

We want to find n such that:

$T_n = 247$ Substituting what we know:

$247 = -5 + (n - 1)3$ Adding 5 to both sides:

$252 = (n - 1)3$ Dividing both sides by 3:

$n - 1 = 84$ Therefore: $n = 85$ Thus, the 85th term in the sequence has a value of 247.

Step 4

Find S_{50}, the sum of the first 50 terms of the sequence.

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Answer

To find the sum of the first n terms of an arithmetic sequence, we use the formula:

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

$n = 50$ (number of terms)
$a = 4$ (first term)
$d = 5$ (common difference found from 4 to 9 and 9 to 14)

Plugging the values into the formula:

$S_{50} = \frac{50}{2} (2 \cdot 4 + (50 - 1)5)$ $S_{50} = 25 (8 + 245)$ $S_{50} = 25 \cdot 253$ Therefore, performing the multiplication: $S_{50} = 6325$

The first three terms of an arithmetic sequence are −5, k, 1 - Leaving Cert Mathematics - Question 6 - 2021

Worked Solution & Example Answer:The first three terms of an arithmetic sequence are −5, k, 1 - Leaving Cert Mathematics - Question 6 - 2021

Find k and hence show that the common difference is 3.

Find T_{10}, the 10th term in the sequence.

Find which term in the sequence has a value of 247.

Find S_{50}, the sum of the first 50 terms of the sequence.

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