JEE Main 2020 (8 Jan Evening)

LEVELBoard

Animated Solution for Mathematics - Sequence and Series: If the $1 0^{t h}$ term of an A.P. is $\frac{1}{20}$ and its $2 0^{t h}$ term is $\frac{1}{10}$ , then the sum of its first 200 term is

Select Answer:

Visualized Solution

Define the A.P. Parameters

Let the first term of the A.P. be $a$ .
Let the common difference be $d$ .

Formulate Equation for $1 0^{t h}$ Term

Given $T_{10} = \frac{1}{20}$
Using formula $T_{n} = a + (n - 1) d$ :
$a + 9 d = \frac{1}{20} \dots (1)$

Formulate Equation for $2 0^{t h}$ Term

Given $T_{20} = \frac{1}{10}$
Using formula $T_{n} = a + (n - 1) d$ :
$a + 19 d = \frac{1}{10} \dots (2)$

Subtract Equations to Eliminate $a$

Subtract equation (1) from equation (2):
$(a + 19 d) - (a + 9 d) = \frac{1}{10} - \frac{1}{20}$

Solve for Common Difference $d$

$10 d = \frac{2 - 1}{20} = \frac{1}{20}$
$d = \frac{1}{200}$

Substitute $d$ to Find $a$

Substitute $d = \frac{1}{200}$ into equation (1):
$a + 9 (\frac{1}{200}) = \frac{1}{20}$

Calculate First Term $a$

$a = \frac{1}{20} - \frac{9}{200}$
$a = \frac{10 - 9}{200} = \frac{1}{200}$

State Sum Formula

Sum of first $n$ terms: $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$
Here, $n = 200$ .

Substitute Values into Sum Formula

$S_{200} = \frac{200}{2} [2 (\frac{1}{200}) + (200 - 1) (\frac{1}{200})]$

Simplify the Expression

$S_{200} = 100 [\frac{2}{200} + \frac{199}{200}]$

Final Calculation

$S_{200} = 100 [\frac{201}{200}] = \frac{201}{2}$

Conclusion

$S_{200} = 100 \frac{1}{2}$
The sum of the first 200 terms is $100 \frac{1}{2}$ .

00:00 / 00:00

The Sigma Insight: Arithmetic Progression (A.P.)

Analyzing the Setup

An Arithmetic Progression (A.P.) is a sequence where the difference between any two consecutive terms is constant. We are given the

1 0^{t h}

term as

T_{10} = 1/20

and the

2 0^{t h}

term as

T_{20} = 1/10

Our goal is to determine the sum of the first

200

terms of this sequence.

The Detective Work

To solve this, we define the first term as

a

and the common difference as

d

. The general formula for the

n^{t h}

term is:

T_{n} = a + (n - 1) d

Using our given milestones, we establish a system of two linear equations:

a + 9 d = \frac{1}{20}

a + 19 d = \frac{1}{10}

The Algebraic Dance

To find the values of

a

and

d

, we eliminate

a

by subtracting the first equation from the second:

(a + 19 d) - (a + 9 d) = \frac{1}{10} - \frac{1}{20}

10 d = \frac{1}{20}

d = \frac{1}{200}

Now, substitute

d = 1/200

back into the first equation to solve for

a

a + 9 (\frac{1}{200}) = \frac{1}{20}

a = \frac{10}{200} - \frac{9}{200} = \frac{1}{200}

It is a remarkable result that both the first term

a

and the common difference

d

are equal to

1/200

The Grand Finale

We now calculate the sum of the first

200

terms using the sum formula:

S_{n} = \frac{n}{2} [2 a + (n - 1) d]

Plugging in

n = 200

a = 1/200

, and

d = 1/200

, we obtain:

S_{200} = \frac{200}{2} [2 (\frac{1}{200}) + 199 (\frac{1}{200})]

S_{200} = 100 [\frac{2}{200} + \frac{199}{200}] = 100 [\frac{201}{200}]

Simplifying the expression, we find:

S_{200} = \frac{201}{2} = 100.5

The sum of the first

200

terms is $100 \frac{1}{2}$ or $100.5$ .

Common Traps & FAQs

Wait, why do we reject the negative root here?

In this specific problem, we are dealing with a linear system of equations, not a quadratic one. There are no roots to reject! We are simply solving for two unknowns,

a

and

d

, using linear algebra. If you ever encounter a quadratic in an A.P. problem, remember to check if the common difference

d

or the first term

a

must be positive based on the context of the sequence.

Why can't we just use energy conservation?

Energy conservation is a principle for physics problems involving mechanics. This is a pure mathematics problem about sequences. Here, we use the 'conservation' of the definition of an Arithmetic Progression: the difference between consecutive terms is constant. Stick to the algebraic definitions of

T_{n}

and

S_{n}

Is there a faster way to find

S_{200}

without calculating

a

and

d

explicitly?

That is a brilliant intuition! Yes, for an A.P., the sum can also be written as

S_{n} = \frac{n}{2} (T_{1} + T_{n})

. However, we don't know

T_{1}

(which is

a

) or

T_{200}

yet. Calculating

a

and

d

is the most robust path here. Sometimes, the 'long' way is the safest way to avoid traps.

Animated Solution for Mathematics - Sequence and Series: If the $1 0^{t h}$ term of an A.P. is $\frac{1}{20}$ and its $2 0^{t h}$ term is $\frac{1}{10}$ , then the sum of its first 200 term is

Select Answer:

Visualized Solution

Define the A.P. Parameters

Formulate Equation for $1 0^{t h}$ Term

Formulate Equation for $2 0^{t h}$ Term

Subtract Equations to Eliminate $a$

Solve for Common Difference $d$

Substitute $d$ to Find $a$

Calculate First Term $a$

State Sum Formula

Substitute Values into Sum Formula

Simplify the Expression

Final Calculation

Conclusion

The Sigma Insight: Arithmetic Progression (A.P.)

Analyzing the Setup

The Detective Work

The Algebraic Dance

The Grand Finale

Similar Questions

If the $1 0^{t h}$ term of an A.P. is $1/20$ and its $2 0^{t h}$ term is $1/10$ , then the sum of its first 200 terms is:

Let $2^{n d}, 8^{t h}$ and $4 4^{t h}$ terms of a non-constant A.P. be respectively the $1^{s t}, 2^{n d}$ and $3^{r d}$ terms of G.P. If the first term of A.P. is 1 then the sum of first 20 terms is equal to-

Let $s_{1}, s_{2}, s_{3}, \dots, s_{10}$ respectively be the sum of 12 terms of 10 A.Ps whose first terms are $1, 2, 3, \dots, 10$ and the common differences are $1, 3, 5, \dots, 19$ respectively. Then $\sum_{i = 1}^{10} s_{i}$ is equal to

If $a_{1}, a_{2}, a_{3}, \dots$ are in A.P. such that $a_{1} + a_{7} + a_{16} = 40$ , then the sum of the first 15 terms of this A.P. is :

Let $T_{r}$ be the rth term of an A.P. If for some m, $T_{m} = \frac{1}{25}$ and $T_{25} = \frac{1}{20}$ , then $5 m \sum_{r = m}^{2 m} T_{r}$ is equal to Note: $20 \sum_{r = 1}^{25} T_{r} = 13$ is given condition in Hindi version.

Suppose $a_{1}, a_{2}, \dots, a_{n}, \dots$ be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms to the sum of first nine terms of the progression is $5 : 17$ and $110 < a_{15} < 120$ , then the sum of the first ten terms of the progression is equal to -

Let $T_{r}$ be the $r$ th term of an A.P., for $r = 1, 2, 3, \dots$ . If for some positive integers $m, n$ we have $T_{m} = 1/ n$ and $T_{n} = 1/ m$ , then $T_{mn}$ equals

Different A.P.'s are constructed with the first term 100, the last term 199, and integral common differences. The sum of the common differences of all such A.P.'s having at least 3 terms and at most 33 terms is ________.

If the sum of the first 40 terms of the series, $3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + .....$ is $(102) m$ , then $m$ is equal to:

Animated Solution for Mathematics - Sequence and Series: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the 10th term of an A.P. is 201​ and its 20th term is 101​, then the sum of its first 200 term is

Select Answer:

Visualized Solution

Define the A.P. Parameters

Formulate Equation for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 10th Term

Formulate Equation for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 20th Term

Subtract Equations to Eliminate .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a

Solve for Common Difference .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } d

Substitute .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } d to Find .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a

Calculate First Term .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a

State Sum Formula

Substitute Values into Sum Formula

Simplify the Expression

Final Calculation

Conclusion

The Sigma Insight: Arithmetic Progression (A.P.)

Analyzing the Setup

The Detective Work

The Algebraic Dance

The Grand Finale

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the 10th term of an A.P. is 1/20 and its 20th term is 1/10, then the sum of its first 200 terms is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let 2nd,8th and 44th terms of a non-constant A.P. be respectively the 1st,2nd and 3rd terms of G.P. If the first term of A.P. is 1 then the sum of first 20 terms is equal to-

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a1​,a2​,a3​,… are in A.P. such that a1​+a7​+a16​=40, then the sum of the first 15 terms of this A.P. is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let Tr​ be the rth term of an A.P., for r=1,2,3,…. If for some positive integers m,n we have Tm​=1/n and Tn​=1/m, then Tmn​ equals

Different A.P.'s are constructed with the first term 100, the last term 199, and integral common differences. The sum of the common differences of all such A.P.'s having at least 3 terms and at most 33 terms is ________.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the sum of the first 40 terms of the series, 3+4+8+9+13+14+18+19+.....is (102)m, then m is equal to:

Animated Solution for Mathematics - Sequence and Series: If the $1 0^{t h}$ term of an A.P. is $\frac{1}{20}$ and its $2 0^{t h}$ term is $\frac{1}{10}$ , then the sum of its first 200 term is

Formulate Equation for $1 0^{t h}$ Term

Formulate Equation for $2 0^{t h}$ Term

Subtract Equations to Eliminate $a$

Solve for Common Difference $d$

Substitute $d$ to Find $a$

Calculate First Term $a$

If the $1 0^{t h}$ term of an A.P. is $1/20$ and its $2 0^{t h}$ term is $1/10$ , then the sum of its first 200 terms is:

Let $2^{n d}, 8^{t h}$ and $4 4^{t h}$ terms of a non-constant A.P. be respectively the $1^{s t}, 2^{n d}$ and $3^{r d}$ terms of G.P. If the first term of A.P. is 1 then the sum of first 20 terms is equal to-

If $a_{1}, a_{2}, a_{3}, \dots$ are in A.P. such that $a_{1} + a_{7} + a_{16} = 40$ , then the sum of the first 15 terms of this A.P. is :

Let $T_{r}$ be the $r$ th term of an A.P., for $r = 1, 2, 3, \dots$ . If for some positive integers $m, n$ we have $T_{m} = 1/ n$ and $T_{n} = 1/ m$ , then $T_{mn}$ equals

If the sum of the first 40 terms of the series, $3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + .....$ is $(102) m$ , then $m$ is equal to: