Find (i) the 22 nd term of the progression 4,9,14,19 ………(ii) the first negative term of the prog

Find (i) the 22 nd term of the progression 4,9,14,19 ………(ii)...

Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This fixed amount, known as the common difference, allows one to determine any term in the sequence by using a linear formula. The study of APs involves understanding how sequences grow or decline by uniform increments, which is fundamental in solving term-finding problems and analyzing series behavior.

nth Term of an Arithmetic Progression

The nth term of an arithmetic progression can be explicitly calculated using the formula a_n = a + (n-1)d, where 'a' is the first term and 'd' is the common difference. This concept is crucial because it provides a systematic method for determining both specific terms and characteristics like when a term becomes negative, by setting up and solving inequalities based on this formula.

Geometric Progression

A geometric progression (GP) is a sequence where each term is found by multiplying the previous term by a constant value called the common ratio. This multiplicative process leads to exponential growth or decay in the sequence and is essential in understanding sequences that change by fixed percentages. The structure of GPs is used to solve problems where terms increase or decrease rapidly and to derive properties involving exponential equations.

nth Term of a Geometric Progression

The nth term of a geometric progression is given by the formula a_n = a × r^(n-1), where 'a' is the first term and 'r' is the common ratio. This formula is central to analyzing GPs as it connects the multiplicative nature of the sequence to its exponential form, enabling the determination of any term and the derivation of related parameters by relating different term positions.

Transcript

00:01 So here we have a series which is given that is 4 914 and 19 these are the first four terms of that series and we are asked to find the 20 second term of this series.

00:13 Okay.

00:14 So first of all we need to identify which kind of series is this that or what kind of progression do we have here.

00:20 So if you observe, you can see that all these terms have the same difference between like all the consecutive terms how the same.

00:30 Difference that is here 9 minus 4 is equal to 5 right also 14 minus 9 is equal to 5 and 19 minus 14 is also equal to 5 that means we have a common difference which is equal to 5 hence we can say that this isn't arithmetic progression right so for an arithmetic progression for a series in arithmetic progression we can say that nth term is given by by where n -th term is tn okay which is given by a plus n minus 1 multiplied by d where a is the first term here in this case 4 and n is the number of terms that is for example the n -th term number which whichever term you want and d is the common difference so here we need to find the 22nd term so for that we just need to substitute n as 22 so t -22 will be 4 plus n minus 1 that is 22 minus 1 multiplied by d that is 5 right so this will be 4 plus 22 minus 1 is 21 21 multiplied by 5 so this will be 4 plus 1 05 which is 1 .9 so that means the 22 term is 109 okay so if that is again then to the second question or the second sub question we have in the second sub question we have another progression that is 19 18 1 by 5 then 17 1 by 5 so these are the three terms so what they've asked is that is the first negative term of this progression so that means t n which is less than 0 is what we need to find out okay so so in that case, first of all, we need to find like, we need to mean if this is an systematic progress for that we need to know if there is a common difference between these three terms, so three terms.

02:47 So first of all like 18, 1 by 5 minus 19 is equal to how much 18 multiplied by 5 plus 1 1 divided by 5 minus 19 right and this will be so we 18 multiplied by 5 is 90 so this becomes 91 by 5 minus 19 right 91 by 5 minus 9 minus 4 by 5 4 by 5 okay so here we got 4 by 5 as the difference now coming to the difference between these two terms 17 1 sorry this is 17 2 by 5 not 1 by 5 okay 17 2 by 5 minus 18 1 by 5 and this will be 17 multiplied by 5 plus 2 divided by 5 minus 18 multiplied by 5 plus 1 divided by 5 right so in this case this will become 17 multiplied by 5 is 8 85 so 85 plus 2 is 87 by 5 minus 18 multiplied by 5 is 90 so this will become 91 by 5 and this will also give us minus 4 by 5 so we have another series in a 20 progression that is with a common difference of therefore common difference is minus 4 by 5 right common difference is minus 4 by 5 now here also what they asked is so as earlier the end term will be given by nth term will be given by the equation tn is equal to a plus n minus 1 into d or n minus 1 multiplied by d where is the first term n is the number of terms and d is the common difference so what they asked is that the first negative turns so we can say that t n is less than or equal to 0 right tiness less than equals 0 so we can say that if t is negative t is less than 0 10 is less than 0 a plus and minus 1 multiple every is also less than 0 so substituting the values we have that will become 19 plus n minus 1 multiplied by d is minus 4 by 5 right this is less than 0 right that is what this so solving this we get 19 19 minus 4 by 5 n plus 4 by 5 less than 0 right and this will become 19 multiplied by 5 multiplied by 5 plus 4 will divided by 5 minus 4 by 5 yeah right and this will be 19 multiplied by 5 is 95 95 plus 95 plus 4 is 99 minus we have an in we already have that 19 19 minus 4n whole divided by 5 is less than 0 and multiplying both sides by 5 we can write that 99 minus 4 n is less than 0 so here we have this so adding 4 n on both sides we can write 99 less than for right or four and greater than 99 which will be the case so that means like dividing both sides like 99 sorry during versus by 4 we'll get n greater than 99 divided by 4 so that means this will be 99 divided by 4 us 24 .75 so that means n is greater than 24 3 by 4 so that means n is equal to and can't be you know and has to be a whole number right not some fraction or anything like that because n is a number right so that means n is equal to 25 so what this means is that any term that comes after you know 24 is a negative term so the 25th term 26 term 27 term all these terms can be negative so the thing is t 25 t25 is the is the first first negative okay so that's what this means t 20 very is the first negative term of the series now to the third question third sub question that is we have another series five plus 10 plus 20 because it goes on.

08:31 So what is the 11th term? what is the 11th term is what they asked.

08:36 So here also we need to find what kind of series is this or what kind of progression is this...

Please give Ace some feedback