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Arithmetic Progression and Geometric Progression | Don't Memorise | (GMAT/GRE/CAT/Bank PO/SSC CGL) - YouTube
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We have seen a sequence like three, five, seven, nine and so on.
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In this sequence three is the first term.
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To define the 'nth' term in a simpler way, we just call the first term as 'a'.
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How do we define the second term then?
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The common difference as we can see is two.
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If we call the common difference as 'd'
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the second term can be written as 'a' plus 'd'.
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What can the third term be written as?
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That's easy!
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We write it as 'a' plus '2d'.
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First term plus twice the common difference.
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So based on this logic, how can we write the 'nth' term?
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We can write it as 'a plus n minus 1 times d.'
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Why 'n minus 1'?
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Observe the pattern, third term '2d',
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second term '1d'.
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So this is the formula we had seen earlier.
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'First term plus n minus 1 times d.'
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It's just that we have written the first term as 'a'.
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Now that we have seen individual terms in a sequence,
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we can get to the sum of their terms.
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How do we find the sum of all the terms in a sequence?
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Think of it in terms of average.
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The average of 'n' terms will be the sum of terms over 'n'.
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So the sum of terms will be 'n' times the average.
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Understanding the average is the best part of it.
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All sequences are evenly spaced numbers.
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For evenly spaced numbers, the average
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is the sum of first term and the last term over 2.
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That's it!
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We get the sum of terms if we know the number of terms.
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The first term and the last term.
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The sum of terms in a sequence is called series.
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We will understand more about it in our future sessions.
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Let's take the 2 outside and write 'n by 2'.
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The first term is 'a' and this is the last term.
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So 'first term plus the last term, is equal to 2a plus n minus 1 times d.'
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This is the formula to find the sum of terms, in a sequence.
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But wait, this is only an arithmetic sequence.
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Yes, this is an arithmetic sequence
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as the difference between two consecutive terms is constant.
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But wait, what are the other types of sequences then?
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To look at more types, we go through the next sequence.
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3, 9, 27,
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81 and so on.
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Pause the video
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and think about whether this is a sequence or not.
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Okay!
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Some of you would have assumed
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that as the difference between two consecutive terms is not constant
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this is probably not a sequence.
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Don't forget the definition of a sequence.
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It is simply a set of numbers which have some pattern.
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And the pattern can be anything.
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Did you spot a pattern here?
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Look at the ratio of two consecutive terms.
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The ratio '9 over 3' is equal to 3.
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And the ratio of '27 to 9' is also 3.
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Just like the common difference in arithmetic,
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we have a common ratio here.
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We can call it 'r'.
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Each term is multiplied by 3 to get the next term.
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This is called a geometric sequence or progression.
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Let's assume the first term of the sequence as 'a'.
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What will be the second term then?
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It will be 'a times' the common ratio 'r'.
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And the third term? It will be 'a times r squared'.
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And what will be the 'nth' term of a geometric progression?
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Look at the pattern,
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'second term r raised to one',
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'third term r raised to two'.
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So the 'nth' term will be 'a times r raised to n minus one'.
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And what will be the sum of the terms in a geometric sequence?
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This is just a formula you need to know.
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'a times 1 minus r raised to n over 1 minus r'.
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The proof of this is easy, and it is good to know.
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But you don't really need to know it for your exams.
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So the four things you should know for the exams?
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'nth term' and 'sum of terms' for an arithmetic sequence.
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And the 'nth' term and 'sum of terms' for a geometric sequence.
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