Mixed Problems: Set 2

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle before we start. I am thinking of three numbers in a G P, a geometric progression. They add up to twenty six, and their squares add up to three hundred and sixty four. Three unknowns, but only two facts. So, can you find them? Looks impossible, right? By the end of this video, you will crack it in a few lines. This is Set two of the mixed problems on sequences and series. In a mixed set, nobody tells you the chapter, so spotting the right tool is half the job. And problems like these come up often in JEE Main. So, let's begin!

0:39Warm-up: an A.P. meets a G.P.

Let's warm up with one from Set two, marked medium. The letters a, b, c are in A P, and x, y, z are in G P, all positive. Find x to the power b minus c, times y to the power c minus a, times z to the power a minus b. The small letters up top are the powers, okay? Before any algebra, what does each progression give you? Very good! For the A P, call the common difference d. Then b minus c is minus d, c minus a is two d, and a minus b is minus d again. With me? And for the G P, y squared equals x times z: the small two up top means squared. The middle term squared is the product of its neighbours. Remember this fact, because it comes back today. Right? Now put in the powers: x to the power minus d, y to the power two d, z to the power minus d. Together, that is y squared over x z, all to the power d. But y squared over x z is one, so the answer is one! And the word positive in the question keeps all these powers well defined. Clear?

1:50The Chapter 4 tool: telescoping

Now, before our two main examples, one tool from Chapter four. Say each term, T k, capital T with a small k below, is f of k, minus f of, k minus one. Then the sum, the big Greek letter sigma, from k equals one to n, is just f of n minus f of zero. Everything in the middle cancels, like a telescope folding shut. Okay? The simplest case: one over k times k plus one equals one over k, minus one over k plus one. Right? And with three factors: one over k times k plus one times k plus two, equals one half times, one over k times k plus one, minus one over k plus one times k plus two. But where does that one half come from? Subtract the two fractions, and the top becomes k plus two, minus k, which is two. The last factor minus the first factor! So we divide by two. Keep this in mind; we need it very soon. Got it?

2:49Example 1: GP numbers with sum 26 and sum of squares 364

Now, remember our puzzle? Let's finish it. Example one, from Set two, marked medium. Three numbers in G P have sum twenty six, and the sum of their squares is three hundred and sixty four. Find them. Pause the video, and try it. Done? Okay, let's check. Call the numbers a, b, c, in order. The identity that links a sum with a sum of squares is the square of a sum: a plus b plus c, the whole squared, equals a squared plus b squared plus c squared, plus two times, a b plus b c plus c a. Okay? Now, the sum is twenty six, and twenty six squared is six hundred and seventy six. So a b plus b c plus c a is six hundred and seventy six minus three hundred and sixty four: three hundred and twelve, right? No, no, careful! Six hundred and seventy six minus three hundred and sixty four is two times, a b plus b c plus c a. The double arrow means which gives: halve it, and a b plus b c plus c a is one hundred and fifty six. Forgetting that two is a very common slip. Clear? Now the G P, and here is the hidden third fact! Middle term squared equals the product of the outer two: a c equals b squared. So replace c a with b squared, and take b common: b times, a plus b plus c. That is twenty six b. With me? So twenty six b equals one hundred and fifty six, and b is six. Right? Then a plus c is twenty six minus six, which is twenty, and a c is six squared, thirty six. Two numbers with sum twenty and product thirty six are the roots of t squared minus twenty t plus thirty six equals zero. So t is two or eighteen. Okay? So the numbers are two, six and eighteen. Check: two plus six plus eighteen is twenty six, and four plus thirty six plus three hundred and twenty four is three hundred and sixty four. Perfect! Three unknowns, two facts, and the G P gave us the third. Puzzle solved!

4:55Example 2: sum of 1/k(k+1)(k+2)(k+3) by telescoping

Now the toughest one in Set two, marked hard. Find the sum, sigma from k equals one to n, of one over k, times k plus one, times k plus two, times k plus three. Four factors in a row this time. Pause the video, and try it. Hint: use the tool we just saw. Done? Okay, let's check. With three factors, we had one half in front. Now there are four factors, so one quarter in front, right? Oh no, careful! Let's actually subtract. One over k times k plus one times k plus two, minus one over k plus one times k plus two times k plus three. Over the common denominator, the top is k plus three, minus k. That's three, not four! The last factor minus the first factor, remember? So we divide by three. One over the four factors equals one third times the same difference, inside square brackets. Each term is now a difference, so the sum will telescope. Okay? Write the first two terms. The dots between the numbers mean times. k equals one gives one over one times two times three, minus one over two times three times four. k equals two gives one over two times three times four, minus one over three times four times five. See? One over two times three times four cancels, and the cancelling goes on all the way. With me? Now look at what survives. Only the very first piece, one over one times two times three, which is one sixth, and the very last piece, one over n plus one, times n plus two, times n plus three. So the sum is one third times that difference. Clear? Open the bracket: one over eighteen, minus one over three times, n plus one, times n plus two, times n plus three. That's the answer! Quick check with n equals one. One over eighteen minus one over seventy two is one over twenty four. And the first term itself is one over one times two times three times four: one over twenty four. Perfect!

7:04Quick revision: sum of squares, GP middle term, telescoping

So, quick revision. Sum and sum of squares given? Square the sum, subtract the sum of squares, and don't forget to halve. Three terms in a G P? The middle term squared is the product of its neighbours, a c equals b squared: the hidden extra fact. Telescoping with factors in a row? Put one over, last factor minus first factor, in front, and let everything cancel. Now go and try the four practice problems of Set two on this page. Spot the tool first, and the rest is easy. All the best!