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## SAT Arithmetic Progression : Terms common to two arithmetic sequence

### Question 4: 18th term common to 2 AP

The first term of an arithmetic sequence is 24 and its common difference is 4. The first term of another arithmetic sequence is 35 and its common difference is 7. What is the 18 term common to both the sequences?

1. 476
2. 56
3. 182
4. 532
5. 142
Choice D. The 18th term common to the two sequences is 532.

This SAT Quant question from sequences and series is an interesting one – you will have to find the 18th term common to both the arithmetic sequences. The level of difficulty is moderate. Use the Hints/Clue given in the first tab to solve the question. If you need more help, click on the subsequent tabs to get a detailed explanation.

1. Find the first term common to both the arithmetic sequences. This has to be found out by listing the two sequences. There is no other alternative.
2. When you list down the first three terms common to the two sequences, you will realize that the terms common to both the sequences is also in an AP.
3. Find the common difference 'd' of the terms that are common to both the sequences.
4. Compute the 18th term common to both the sequences.

### Step 1: Find the first term common to the 2 sequences

There is only one way to determine the first term common to both the sequences. List down first few terms of both the sequences and find the terms common to both.

Sequence 1: 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, ....

Sequence 2: 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119,, ....

Terms common to both sequences: 56, 84, 112, ...

First term common to both the arithmetic sequences is 56.

### Step 2: Compute the common difference 'd' of the terms common to both the sequences

The first term common to both the sequences is 56.

The second term common to both the sequences is 84 and the third term common to both the sequences is 112.

So, the terms common to both the arithmetic sequence are also in another arithmetic progression

The common difference of this sequence is 28.

### How does one determine the common difference 'd' of the terms common to both the sequences?

The common difference of the first sequence is 4. i.e., the first sequence moves in steps of 4.

The common difference of the second sequence is 7. Therefore, the second sequence moves in steps of 7.

The meeting point of the two sequences will move in steps that should be common to both - i.e., steps of 4 and steps of 7.

i.e., the common terms will move in steps of 28 - the LCM of 4 and 7.

the common difference of the sequence comprising terms common to both the sequences will be the LCM of the common differences of the two sequences.

### Step 3: Compute the 18th term common to both the sequences

The first term common to both the sequences is 56.

The common difference of the terms common to the two sequences is 28.

18th term of an AP t18 = t1 + (18 - 1)d.

t18 = 56 + 17 * 28.

t18 = 532

### Try this variant yourself

The first term of an arithmetic sequence is 23 and its common difference is 5. The first term of another arithmetic sequence is 19 and its common difference is 7. What is the 7th term common to both the sequences?

1. 210
2. 243
3. 35
4. 33
5. 68

## Concept Overview : Arithmetic Progression

Get a quick overview on the basic concepts of Arithmetic sequence including formulae to find the nth term of an arithmetic sequence and the sum of n terms of an arithmetic sequence.

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