Let a1,a2,a3,… be an arithmetic progression with a1=7 and common difference 8. Let T1,T2,T3,… be such that T1=3 and Tn+1-Tn=an fo n≥1. Then, which of the following is/are TRUE?

JEE Advanced · Mathematics · 5. Sequences & Series

Let $a_{1}, a_{2}, a_{3}, \dots$ be an arithmetic progression with $a_{1} = 7$ and common difference $8$ . Let $T_{1}, T_{2}, T_{3}, \dots$ be such that $T_{1} = 3$ and $T_{n + 1} - T_{n} = a_{n}$ fo $n \geq 1$ . Then, which of the following is/are TRUE?

A $T_{20} = 1604$
B $\sum_{k = 1}^{20} T_{k} = 10510$
C $T_{30} = 3454$
D $\sum_{k = 1}^{30} T_{k} = 35610$

Verified Solution

Answer & Solution

Correct Answer

(B) $\sum_{k = 1}^{20} T_{k} = 10510$

Step-by-step Solution

Detailed explanation

Given,

a_{n} = 7 + (n - 1) 8

and

T_{1} = 3

Also

T_{n + 1} = T_{n} + a_{n}

Now

T_{n} = T_{n - 1} + a_{n - 1}

Putting

n = 1

we get,

T_{2} = T_{1} + a_{1}

Now putting

n = 2

we get,

T_{3} = T_{2} + a_{2} \Rightarrow T_{3} = T_{1} + a_{1} + a_{2}

And so on we get,

\Rightarrow T_{n + 1} = T_{1} + a_{1} + a_{2} + \dots + a_{n}

\Rightarrow T_{n + 1} = T_{1} + \frac{n}{2} [2 (7) + (n - 1) 8]

\Rightarrow T_{n + 1} = T_{1} + n (4 n + 3) \dots (1)

Now for

n = 19

we get,

T_{20} = 3 + (19) (79) = 1504

And for

n = 29

we get,

T_{30} = 3 + (29) (119) = 3454

{option C is correct}
Now finding sum we get,
$\sum_{k=1}^{20} T_k=3+\sum_{k=2}^{20} T_k=3+\sum_{k=1}^{19}(3~+$ $4 n^2+3 n)$

= 3 + 3 (19) + \frac{3 (19) (20)}{2} + \frac{4 (19) (20) (39)}{6}

= 3 + 10507 = 10510

{option B is correct}
And Similarly

\sum_{k = 1}^{30} T_{k} = 3 + \sum_{k = 1}^{29} (4 n^{2} + 3 n + 3) = 35615

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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$(A)$	In a triangle $∆ X Y Z$ let $a, b$ and $c$ be the lengths of the angles $X, Y$ and $Z$ , respectively. If $2 (a^{2} - b^{2}) = c^{2}$ and $λ = \frac{\sin (X - Y)}{sinZ}$ then possible values of $n$ for which $\cos (n π λ) = 0$ is are	$(P)$	$1$
$(B)$	In a triangle $Δ X Y Z,$ let $a, b$ and $c$ be the lengths of the sides opposite to the angles $X, Y$ and $Z$ , respectively. If $1 + \cos 2 x - 2 \cos 2 y$ $= 2 \sin x \sin y$ , then possible value(s) of $\frac{a}{b}$ is (are)	$(Q)$	$2$
$(C)$	In $R^{2},$ let $\sqrt{3} \hat{i} + \hat{j}, \hat{i} + \sqrt{3} \hat{j}$ and $β \hat{i} + (1 - β) \hat{j}$ be the position vectors of $X, Y$ and $Z$ with respect to the origin $O$ , respectively. If the distance of $Z$ from the bisector of the acute angle of $\vec{O X}$ with $\vec{O Y}$ is $\frac{3}{\sqrt{2}}$ , then possible value (s) of $\|β\|$ is (are)	$(R)$	$3$
$(D)$	Suppose that $F (α)$ denotes the area of the region bounded by $x = 0, x = 2, y^{2} = 4 x$ and $y = \|a x - 1\| +$ $\|α x - 2\| + α x,$ where $α \in \{0, 1\} .$ Then the value (s) of $F (α) + \frac{8}{3} \sqrt{2},$ when $α = 0$ and $α = 1$ , is (are)	$(S)$	$5$
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(B) The set of points $z$ satisfying $\|z+4\|+\|z-4\|=0$ is contained in or equal to	(q) the set of points $z$ satisfying $\operatorname{Im}(z)=0$
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Column I	Column II
(A)The set of points \(z\) satisfying \(\|z-i\| z\|\|=\|z+i\| z\|\|\) is contained in or equal to	(p)an ellipse with eccentricity \(4 / 5\)
(B) The set of points \(z\) satisfying \(\|z+4\|+\|z-4\|=0\) is contained in or equal to	(q) the set of points \(z\) satisfying \(\operatorname{Im}(z)=0\)
(C) If \(\|w\|=2\), then the set of points \(z=w-\frac{1}{w}\) is contained in or equal to	(r) the set of points \(z\) satisfying \(\|\operatorname{Im} z\| \leq 1\)
(D) If \(\|w\|=1\), then the set of points \(z=w+\frac{1}{w}\) is contained in or equal to	(s) the set of points satisfying \(\|\operatorname{Re} z\| \leq 2\)
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